\
LIBRARY
'
STAT.
STAT.
AN INTRODUCTION TO
THE LIE THEORY OF
ONE-PARAMETER GROUPS
WITH APPLICATIONS TO
THE SOLUTION OF DIFFERENTIAL EQUATIONS
BY
ABRAHAM COHEN, PH.D.
ASSOCIATE IN MATHEMATICS, JOHNS HOPKINS UNIVERSITY
D. C. HEATH & CO., PUBLISHERS
BOSTON NEW YORK CHICAGO
II I
BY THE SAME AUTHOR
AN ELEMENTARY TREATISE ON
DIFFERENTIAL EQUATIONS
ix -f- 271 pages. Half Leather
D. C. HEATH & CO.. PUBLISHERS
COPYRIGHT, 1911,
BY ABRAHAM COHEN.
MATH.
STAT.
UBRARY
STAT •
PREFACE
THE object of this book is to present in an elementary manner,
in English, an introduction to Lie's theory of one-parameter groups,
with special reference to its application to the solution of differen
tial equations invariant under such groups.
The treatment is sufficiently elementary to be appreciated, under
proper supervision, by undergraduates in their senior year as well
as by graduates during their first year of study.
While a knowledge of the elementary theory of differential equa
tions is not absolutely essential for understanding the subject
matter of this book, frequent references being made to places where
necessary information can be obtained, it would seem preferable to
approach for the first time the problem of classifying and solving
differential equations by direct, even if miscellaneous, methods to
doing so by the elegant general methods of Lie ; and this book is
intended primarily for those who have some acquaintance with the
elementary theory. To such persons it should prove of great inter
est and undoubted practical value. An attempt has been made
throughout the work to emphasize the role played by the Lie theory
in unifying the elementary theory of differential equations, by
bringing under a relatively small number of heads the various
known classes of differential equations invariant under continuous
groups, and the methods for their solution. Special attention may
be called to the lists of invariant differential equations and applica
tions in §§ 19, 28, 30; while the two tables in the appendix include
most of the ordinary differential equations likely to be met.
Only as many examples involving the solution of differential
equations as seem necessary to illustrate the text have been intro-
iv PREFACE
duced. The large number usually given in the elementary text
books seems ample for practice.
The short chapter on contact transformations, while not essential
to the work, has been added for purposes of reference and to give
the student sufficiently clear ideas, so as to provide a working
knowledge, in case he has occasion to apply them. For the same
reasons, the rather sketchy note on /--parameter groups has been
added, where an attempt is made to bring out, as concisely as
seems consistent with clearness, the relations between r-parameter
groups and their infinitesimal transformations. An exposition of
the general theory would be beyond the scope of this work.
To a large extent Lie's proofs and general mode of presentation
have been retained, both because of their elementary, direct char
acter, and because the subject is so essentially Lie's own. An
attempt has been made, however, at a more systematic arrange
ment of the subject matter and at identifying more closely the
classes of differential equations invariant under known groups with
those considered in the elementary theory.
The author takes pleasure in expressing his appreciation of the
valuable suggestions made by Dr. J. R. Conner, who kindly con
sented to read the proofs.
ABRAHAM COHEN.
JOHNS HOPKINS UNIVERSITY,
BALTIMORE, MD., August, 1911.
CONTENTS
CHAPTER I
LIE'S THEORY OF ONE-PARAMETER GROUPS
SECTION PACK
1. Group of Transformations ......... i
2. Infinitesimal Transformation * 6
3. Symbol of Infinitesimal Transformation 8
4. Group Generated by an Infinitesimal Transformation 10
5. Another Method of Finding the Group from its Infinitesimal Transfor
mation .. ^ ......... 14
6. Invariants 16
7. Path-curves. Invariant Points and Curves 17
8. Invariant Family of Curves 20
9. Change of Variables 23
10. Canonical Form and Variables 26
11. Groups Involving More than Two Variables 28
CHAPTER II
DIFFERENTIAL EQUATIONS OF THE FIRST ORDER
12. Integrating Factor 37
13. Differential Equation Invariant under Extended Group ... 40
14. Alternant 44
15. Another Criterion for Invariance of a Differential Equation under a
Group ............ 45
16. Two Integrating Factors 48
17. General Expression for Group under which a Differential Equation is
Invariant . 49
18. Differential Equations Invariant under a Given Group . ... 50
19. Illustrations and Applications ........ 52
20. Second General Method for Solving a Differential Equation. Separa
tion of Variables 63
21. Singular Solution 66
v
vi CONTENTS
CHAPTER III
MISCELLANEOUS THEOREMS AND GEOMETRICAL APPLICATIONS
SECTION PAGE
22. New Form for Integrating Factor ....... 69
23. Two Differential Equations with Common Integrating Factor . . 72
24. Isothermal Curves .......... 72
25. Further Application of the Theorem of § 23 ..... 76
CHAPTER IV
DIFFERENTIAL EQUATIONS OF THE SECOND AND HIGHER ORDERS
26. Twice-extended, w-tiines-extended Grbup ...... 83
27. Differential Equation of Second Order Invariant under a Given Group 86
28. Illustrations and Applications ........ 90
29. Further Applications .......... 97
30. Differential Equation of Order Higher than the ^econd Invariant under
a Given Group ........... 99
CHAPTER V
LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER
31. Complete System ........... 104
32. Method of Solution of Complete System ...... 1 1 1
33. Second Method of Solution ......... 113
34. Linear Partial Differential Equation Invariant under a Group . 115
35. Method of Solution of Linear Partial Differential Equation Invariant
under a Group ........... 119
36. Jacobi's Identity ......... . .121
37. Linear Partial Differential Equation Invariant under Two Groups „ 122
38. Methods of Solution of Linear Partial Differential Equation Invariant
under Two Distinct Groups ........ 124
CHAPTER VI
ORDINARY DIFFERENTIAL KQI-ATIONS OF THK SITOND ORDER
39. Differential Equation of the Second Order Invariant under a Group . 134
40. Differential Equation of the Second Order Invariant under Two Groups 137
41. Other Methods of Solution ......... 142
CONTENTS vii
SECTION PAGE
42. Number of Linearly Independent Infinitesimal Transformations that
Leave a Differential Equation of the Second Order Unaltered,
Limited 143
43. r-parameter Group of Infinitesimal Transformations .... 146
44. Classification of Two-parameter Groups 152
45. Canonical Forms of Two-parameter Groups 155
46. DiftVrential Equation of the Second Order Invariant under Two Groups 165
47. Second Method of Solution for B 169
CHAPTER VII
CONTACT TRANSFORMATIONS
48. Union of Elements 175
49. Contact Transformation 178
50. Group of Contact Transformations. Infinitesimal Contact Transforma
tion . . • *• • • • • l85
51. Ordinary Differential Equations 189
52. First or Intermediary Integrals 191
53. Differential Equation of the First Order Invariant under a Group of
Contact Transformations . . . . . . . . .194
APPENDIX
Note I. — The Infinitesimal Transformation 197
Note II. — Solution of the Riccati Equation of § 18 201
Note III. — Isothermal Curves 203
Note IV. — Differential Equation of the Second Order not Invariant under
Any Group 206
NoteV. — (t71'C7«')/=(t7iUzy/ ' . .209
Note VI. — Continuous Groups Involving More than One Parameter . .211
Note VII. — Condition for Essential Parameters 226
Table I — Differential Equations of the First Order Invariant under Known
Groups 231
Table II. — Differential Equations of Higher Order Invariant under Known
Groups 236
ANSXVKKS TO EXAMPLES 239
INDEX „ 247
LIE'S THEORY OF
DIFFERENTIAL EQUATIONS
CHAPTER I
LIE'S THEORY OF ONE-PARAMETER GROUPS
1. Group of Transformations. — The set of transformations
(i) xl = <l>(xlyt a), yi = \l/(x,y, a)*
each one being determined by some value of the parameter a, con
stitutes a group if the transformation resulting from the successive per
formance of any two of them is one of the transformations of the
aggregate. In other words, assigning a definite but arbitrarily selected
value to the parameter a, and then any second value b (where b may
or may not be equal to a), this second transformation being
(i6) ,r2 = 4>(xlt ylt &), y, = *l/(xlf y}, 6),
the transformations of type (i) form a group if the results of eliminat
ing Xi and j't from (i) and (i6), i.e.
,yt a), \l/(x,y, a\ ^]» y-2
* Here 0 and \f/ are supposed to be generally analytic, real functions of the three
quantities xty, a; and, unless especially stated, it will be understood that .r and y are
real, and that a takes such values only as render .rj and y^ real. Besides, 0 and \J/ are
independent functions with respect to .r anfi y, alone; i.e.
(90 (>/>
_r dy
so that equations (i) can be solved for .1 and y.
2 THEORY OF DIFFERENTIAL EQUATIONS §i
reduce identically to
x2 = <j>(x, y, c), y2 = $(x, y, c\
where c is a function of a and b only. If (i) be represented by Tn
and (i6) by Tb, the group property may be expressed symbolically
TaTb=Tc.
We shall speak of TaTb as the product of Ta and Tb ; and shall under
stand that it represents the transformation resulting from the succes
sive performance of Ta and Tb, in the order named. With this in
mind, the group property of a set of transformations may be expressed
in the words, the product of any two transformations of the group is
equal to some transformation of the aggregate.
As an example, consider the translations*
I x\ — x, yi — y-\r a.
After having fixed upon some value a of the parameter, a second transformation
of the set, corresponding to the value l>, is
•*a = *\i )'-2 =-y\ + b-
The result of the successive performance of the two is
x2 = x, y* = y + a + l>,
which is again a translation of the set, with a -f /> as the value of the parameter.
Hence, all translations of the type I form a group.
As another example, consider the rotations^
II Xi = x cos a — y sin a, y\ = x sin a + y cos a.
* It will frequently be found convenient to consider this subject from a geometrical
point of view. A transformation of the form (i) may be looked upon as transforming
the point (x,y) into the point (TJ,;',). The effect of a transformation I is, obviously,
to carry any point the distance ,i in the direction of the axis of r. So that the effect on
all the points of the plane is that of a translation of the whole plane over a distance a
in the dire'etion of the axis of v.
f Obviously the effect of a transformation of this type on the various points of the
plane is that of a rotation of the whole plane, through the angle, a, about the origin.
§i THKORY OK <>NTK- PARAMETER GROUPS 3
The result of first performing the transformation corresponding to some definite
value of a, and then a second one,
x-z = xi cos b — yi sin b, y* = xi sin b -j- y\ cos b,
is x2 = x cos (a + /;) —7 sin (a + £), y» = x sin (a + /;) +y cos (<i + <$),
which is again a rotation of the set, with a + ^ as the value of the parameter.
Hence, all rotations of the type II form a group.
The ajfine* transformations,
III xi =x, yi= ay,
form a group, since the result of two transformations in which the values of the
parameter are a and l>, respectively, is
•*2 = •*» y* — a h',
where ab is the value of the parameter.
In the same way it is readily seen that the perspective or similitudinous f
transformations,
IV . xi = ax, yi = ay,
form a group.
In the groups considered in the Lie theory it is presupposed that
the transformations can be arranged in pairs, the members of which
are mutually inverse \ ; that is, if (i) be solved for x and y, their
values in terms of x^ and y\ assume the forms
(i) x = <K*i, y\, a), y = «AOi> y\> «)>
where a is some function of a.
Thus in the examples above we have the inverse transformations :
I. x = xi, y=yl — a; here a =— a.
II. x = xi cos a •}- y\ sin a, y = — x\ sin a + y\ cos a ; a = — a.
III. x = x\, y = ~y-l-i
a
iv. * = !jrlf y = *y\\ * = -
a a a
* Following Lie, this name is used here in a restricted sms<> to npply to transforma
tions of the types III and III', $ 19. The term goes back to Moebius (1790-1868), and
usually includes all entire linear transformations Xi =<Zj x + b\y -\-Ci,yi~<i-2%~\~ b^y + ^3.
f So called because the effect of any one of them is to stretch the vector going from
the origin to the point (.r,_y) in the ratio (I , leaving its direction unaltered. Any figure
I
in the plane is, therefore, transformed into one similar to it by a transformation IV.
J Such groups will be referred to as Lie groups when this property is to be em
phasized.
4 THEORY OF DIF1- KRKN TIAL EQUATIONS § i
Since the successive performance of two mutually inverse trans
formations results in the identical* transformation, the latter must
always be a transformation in every group considered in this theory f ;
hence, there must always exist a value, a(), of the parameter which
reduces the corresponding transformation to an identity
,) a0)=x,
It is readily seen that in the case of I, II, III, IV the values of «0 are o, o, I, I
respectively.
Since <£ and if/ are continuous functions of the parameter a, if we
start with the value aQ, and allow a to vary continuously, the effect
of the corresponding transformations on x and y will be to transform
them continuously ; that is, for a sufficiently small change in a the
changes in x and y are as small as one pleases. Looked at geometri
cally, the effect will be to transform the point (x, y) to the various
points on some curve, which is known as a path-curve of the group.
Thus in the case of I, the point (x, y) is transformed into the various points
on the line through it, parallel to the axis of y\ in the case of II, the path-curves
are obviously circles having the origin for center ; in III the path-curves are
again lines parallel to the axis of y, while in IV the path-curves are straight lines
through the origin.
It is evident that when x and y are considered as constants while
xl and yl are taken as variables, the equations (i) are the parametric
equations of the path-curve through the fixed point (x, y). Hence,
the path-cuwe corresponding to any point (x, y) may be obtained by
eliminating a from the two equations of (\).
* Identical transformation is the name given to a transformation that leaves un
altered all the elements upon which it operates.
\ Groups exist in which the parameter enters in such a way that there is no iden
tical transformation. (S.-«- Lie, I'ran.<;f<>ni/ati(»i\!;riif>pen,Vo\.\<{4<\.) Such groups
will not be included among those considered here.
§i THEORY OF ONE-PARAMETER GROUPS 5
Remark i. — It is readily seen that, in general, the path-curve
corresponding to any point corresponds equally well to every other
point on it.
There is a possible exception to this statement. A point may be left un
altered by every transformation of the group; as, for example, the origin in the
case of II. Such a point would naturally not have a path-curve. In the case of
III, every point on the axis of x is left unaltered; hence, a line parallel to the
axis of y is the path-curve of every point on it, except the point where it cuts the
axis of x. In IV a line through the origin is the path-curve of every point on it,
except the origin, which is left unaltered.
Remark 2. — The parameter may appear in various forms in the
transformations that determine a given group.
Thus x\ — x, y\ —y + a1 also determines the group of translations I. In this
case a must take imaginary values, as well as real ones, in order to give all the
transformations of I. As a matter of fact a = ia. On the other hand, a negative
value for a determines the same transformation as the corresponding positive
value.
The group of rotations II can also obviously be written
x\ — x Vi — a1 — ya, y\ — xa + yVi — a1.
It is always possible (and in an indefinite number of ways) to
choose as a new parameter such a function of the parameter appear
ing in any group that the value giving the identical transforma
tion is any desired number. For example, this number will be 0 if
a is replaced by a^e'1'0. In particular it will be zero if a is replaced
by a«e".
«
Thus if III and IV, where at) = I, are written
•*"! = x, y\ — eay and x\ — e°x, y\ = e*y,
respectively, a = o will determine the identical transformation. In this form,
complex values of a are necessary to determine transformations which cor
respond to negative values of the parameter in the original forms of the trans
formations of these groups.
6 THEORY OF DIFFERENTIAL EQUATION'S §§ I, 2
Show that the following transformations constitute a group. Find
the respective values of the parameter that give both the inverse and
the identical transformations. Also find the path-curves : —
Ex. 1. xi = ax, Vi = -y. Ex. 2. x{ — a~x, }\ = ay.
Ex. 3. A\ = a'2x, )\ = a*y.
Ex. 4. xl = + .v" + 2 a, }\ = -\- vV — a.
Ex. 5. A*! = x cosh a +y sinh a,yi = xs\nha -\~y cosh a.
Ex. 6. ^,=
i — ax ' i — ax
Ex. 7. .T! = ax-\-(a— i)j', ,i'i=J.
Ex. 8. #! = ^a(.r cos d! — y sin «),' ji = e\x S\T\a-\-y cos a).
2. Infinitesimal Transformation. — Since ^> and ^ are continuous
functions, the transformation
x\ = 4>(x, y, 0o + 8«), y\ = ^(x, y, «0 + &?),
where AO is the value of the parameter determining the identical trans
formation and &a is an infinitesimal, changes x and y by infinitesimal
amounts. Developing by Taylor's Theorem
Noting that <#>(.r,>', «0)=^ ^(^^» ^o) = J, the chang'es in x and
due to the transformation are
§2 THEORY OF ONE-PARAMETER GROUPS /
where terms in higher powers of $a are indicated by dots. Since aQ
is a fixed value of the parameter, the only variables remaining in
and f^\ are x and y. Writing
>•
the transformation takes the form
Higher powers of the infinitesimal 8a may be neglected, provided at
least one of £ and 77 does not vanish identically (i.e. for all values of
x and y), and neither of them is infinite. In this case the transforma
tion producing an infinitesimal change in the variables is
(2) 8* = £(.*,jO&i, Sy = r,(x,y)?>a.
This is known as an infinitesimal transformation.
Remark i. — Sinoe k$a, where k is any finite constant different
from zero, is an infinitesimal when £z is, the latter may be replaced
by the former in (2). Hence, the infinitesimal transformation (2) is
the same as n, ,«./
On -the other hand, if/(.r, v) is not a constant,
&r = /(*, y) • ^, y)Sa, 8v =/(.v, v) - ^
is distinct from (2).
Remark 2. — In case [ *] and f ^] are both identically zero,
or if one of them is infinite, the method of this section for finding an
infinitesimal transformation of the group must be modified. In Note I
of the Appendix the existence of an infinitesimal transformation of
the group is established in every case, and a method for finding it is
also given. Moreover, in the same note it is proved t\\-\\ a one -param
eter group contains only one Distinct infinitesimal transformation.
8 THEORY OF DIFFERENTIAL EQUATIONS §§ 2, 3
In general, the method of this section for finding £ and 77 will be
found applicable ; when not, that of Note I may be employed.
In the case of I the infinitesimal transformation is
•*i = x, y\ = y + 8a,
or dx = o, dy = 8a ;
f=o, ,=,. r3*=ftf*=,.i
Lda da J
For II, the infinitesimal transformation is
x\ = x cos (da) — y sin (da), y\.—x sin (da} + y cos (da}.
Since cos(5«) = i - + ..., and sin (5«) = $a — + ..., and infini
tesimals of higher order than the first may be neglected, cos(Srf) may be re
placed by i, and sin (8a) by da. Hence,
dx = — yda, dy = xda ;
«— * '-*
Similarly, it is readily seen that for
III £ = o, i=y,
IV l = x, r)=y.
Ex. Find the infinitesimal transformations of the groups in the
exercises of § i.
3. Symbol of Infinitesimal Transformation. — In the infinitesimal
transformation
(2) &x = t(x,y)Ba, 8y = y(x, y)Ba,
8 is the symbol for differentiation with respect to the parameter a ;
but in a restricted sense, since it is used to designate the value
which the differential of the new variable x^ or \\ assumes when a =
a0* Thus
* The exceptional cases noted in Remark 2, $ 2 are due to the way in which the
parameter enters and are not peculiar to any .qrmip. (See § 4.) Hence.no modifica
tion of the statement made in the text need be insisted upon, provided it is understood
that the parameter is chosen in proper form.
§3 THEORY OF ONE-PARAMETER GROUPS 9
If/(jc, y) is a generally analytic function of x and y, the effect
of the infinitesimal transformation on it is to replace it by
/(# + £&*, y+rj&a), which on expanding by Taylor's Theorem
becomes
x dy
Hence, y
Lie introduced the very convenient symbol 6^ for the coefficient
of Ba in this expansion ; so that
where
(3) #S«
It is readily seen that 6^= /
where A =-/(xlf yj .
In particular C^c = ^, CJ' = >;.
Since Uf can be written when the infinitesimal transformation (2)
is known, and conversely, (2) is known when Uf is given, Uf is said
to represent (2). For convenience of language we shall usually
speak of " the infinitesimal transformation Uf" instead of " the trans
formation represented by Uf" But it must be borne in mind that
Uf is nota. transformation; it is only the representative of one.
The infinitesimal transformations in the cases of I, II, III, IV are
respectively.
Remark. — The differential operator £7= £ 1- -n — has striking
av or
properties, many of which will be brought out in the course of this
10 THEORY OF DIFFERENTIAL EQUATIONS §§3,4
work. It is, to a large extent, because of these properties that Lie's
introduction of the idea of the infinitesimal transformation has
proven so prolific of results.
4. Group Generated by an Infinitesimal Transformation. — In § 2
was given a method for finding the infinitesimal transformation of a
one-parameter group when the finite transformations are given. Con
versely, the finite transformations can be obtained when the infini
tesimal transformation is known.
Attention was called in Remark 2, § i, to the fact that the
parameter may be made to enter in such a way that the identical
transformation is given by any desired value of the parameter. It
is frequently convenient to have the parameter in such a form
that its vanishing gives the identical transformation. In future,
when this is specifically understood, / will be used for the param
eter. In the general case, when this form is not insisted upon, a
will be retained.
The infinitesimal transformation £7=£ — + n — , or
dx dy
(2') &r = ^,^)8/, 8j' = ^,j-)8/,
carries the point (.r, r) to the neighboring position (x -\- &t,
y + rj8/). The repetition of this transformation an indefinite
number of times has the effect of carrying the point along a path *
which is precisely that integral curve of the system of differential
equations
(4) 5 = <(*,,J',), f = *<•*«•.>•')•
which passes through the point (x, /). At any stage of the above
process x and y have been transformed into xl and jr,, and the
* This is obviously tin- path-curve (^ i) of the group, corresponding to the point
(*,>).
§4 T1IKORY OK ONE-PARAMETER GROUPS II
formulae of transformation are given by those solutions of (4) or of
their equivalents
for which Xi reduces to x and y\ to j for /= o.
The first two members of (5) being free of / form a differential
equation whose solution may be written
since x\ = x, y\=y when /=o. This is the equation of the path-
curve corresponding to the point (xt y).
Solving ufa, }>i) = c for one of the variables,* to fix the idea, say
jfl = o>(v1, c\ and replacing xl in 77 by to, the resulting differential
equation »,
^- = dt
can be solved by a quadrature. Replacing c by its value in terms of
xl and )\ this solution takes the form
v(xlt }\) - t = const. = v (x, y).
Hence it follows that
(6)
determine x^ and y\ as those solutions of (4) or (5) which reduce to
x and y respectively for / = o.
Looking upon (6) as a transformation, the following may be noted :
i° The result of the successive performance of two transformations
U(xlt _v,) = //(.v, v) 1 | //(.v,, v,") : ;/(.v,. r, ),
\ v(X'>, y->)= yu-,. .r, > 4- /',
* At limes it \\ill l>e more practical to uso somt> of tlie other methods given in
the author's Klrmrnt;,! v Treatise on I Mftcivntial I '.(piations (in future referred to as
Kl. 1 >it. Eq.) § 65 for finding a second solution of (5).
12 THEORY OF DIFFERENTIAL EQUATIONS §4
is the same as that of the single transformation
2° The value — / determines the transformation inverse to that
obtained by using /.
3° t=o gives the identical transformation.
Hence the aggregate of all the transformations (6) for all values
of / constitute a group of the kind considered in the Lie theory ( §i).
This group (either in the form (6) or when solved for xl and j,) is
known as the group generated by the infinitesimal transformation (2').*
Moreover, the parameter enters in such a way that f l ) = £(x, y),
~ \ = r^x, y). Since there was no restriction placed on the £ and
77 in (2'), other than that they are generally analytic, which is always
presupposed, we have shown that it is always possible to put the finite
transformations of a one-parameter group in such form that the ex
ceptional cases noted in Remark 2, § 2 will not arise.
dx\ dv\ dt
In 1. equations ( O are — = -^— - =. — •
o i I
v(*\>y\)=y\ -y + '•
In II, dxL = </y1 = df
- y\ *\ i
. •. «Oi, yi) = xi2 + }>i* = x'~ -f y2.
Using method 3° (a) of § 65, El. Dif. Eg.
• '• v(x,y)=tan'}- ' = tan"1" -\- t.
X | .V
* Since the finite transformations of a group can be calculated when its infinitesi
mal transformation is known, the latter may be looked upon ;is the representative of
the Ljroup. We shall often speak of " the group ITf" understanding by this " the group
whose infinitesimal transformation i^ lepresented by Uf."
§4 TIIKORY OF ONE-PARAMETER GROUPS 13
To solve these two equations for *i and y\, so as to obtain the transformation
in the usual form, one may proceed as follows :
Taking the tangent of each side of the second relation,
i'i y 4- .YT
— = — — , where r = tan /.
xi x — yr
Adding I to the square of each side and taking account of the first relation,
Whence x\ = — — = xcost — y sin /,
Vl+T*
— = cos / and =_ = sin /;
Vi + T2 Vi + T'<
XT + V
and Vi = — = = x sin / + y cos /.
Vi + T'2
In III, it is readily seen that
*(JT1, /l) =Xi = X,
v(*i, vi) =\ogyi = log^ + /, oryi = *y.
Note. — It is evident that the solutions of (5) need not always be found in the
form (6). Other forms may be easier to solve for x\ and y\. Thus in IV
log x\ — log x + I and log^i = log y + /
are a pair of obvious solutions of the differential equations (5), and lead at once to
*\ = *xt y\ = <*y.
Find the groups whose infinitesimal transformatiohs are the follow
ing :
Ex. 1. 4^-"r- Rx- 5- *¥ + *¥•
ox " oy ox oy
Ex.2. 2.1-f+jf. Ex.6, .v-'f + .vrf.
3.v dy dx . - dy
Ex.3. 2.rg+3J.|. Ex.7. (.v+.v)f-
Ex. 4. 'f-^f- Ex.8. (,-,.)f+(.v+>4.
.\ (U- 2V dy *'&» - dy
14 THEORY OF DIFI'ERENTIAL EQUATIONS §5
5. Another Method of Finding the Group from its Infinitesimal
Transformation. — Startin with an infinitesimal transformation
dy
it was seen in § 4 that the finite transformations of the group
(i') *i = <l>(x,y, t),y\ = t(x,y, /)
generated by it can be found in such form that
The finite transformations can be obtained (expanded in powers
of /) without integration by means of the following considerations :
The effect of any transformation (V) being to replace x and y by
xl and yly it will change any function /(x, y) into /(xlt }\). Assum
ing/^, j) to be generally analytic, since f(xlf ji) depends upon / it
can be developed by Maclaurin's Theorem.
where f=f(x, y),A =/(»i, 2/i). Writing likewise
so that (^)0 = £ (^o = 77, ( C/j/Oo = Uf, it follows that
=UJ, whence W
Moreover = ^7, = ^ (7^= U&.
Hence f^A = UUf=
Similarly = UUUf=k U*f; and so on. Hence the effect of
§5 THEORY OF ONE-PARAMETER GROUPS
any finite transformation (V) on /is given by
(7) /i
In particular the finite transformations of the group are given by
the formula (7) when /is simply x and y, thus
t*
(8) *^\
t
2 I
where, it will be recalled (§ 3), Ux = £, Uy = rj.
It is readily seen that for the group in the form (8) as in the form
In I Uf=&.
Ux = o, U^x - o, ••• ; Uy = i, U~y — o, U'Ay = o, ....
Hence x\ = x, y\ = y -f- /.
In II Uf=-y&+x&
Qjc {jy
Ux = -y, U*x = U(-y} =- x, U*x= U (- x)= yt and so on.
_ .<! + '1 _...)_>(,_/! + i _...)
2! 4! / V 3! 51 /
Hence
— X COS / —
Similarly y± =
= x sin / -f y cos /.
In III Uf=y §/-m
dy
Ux — o, U*x - o, ... ; Uy -y, U*y = y, U*y
Hence x\. — x, y\_ = y( i + / + /- -|- r f ...)
'V 2! 3! /
* Symbolically this may be written /", _ tf('f.
l6 THEORY OF DIFFERENTIAL EQUATIONS §§5,6
In exactly the same v ay the finite transformations of IV are found to be
xi = xel,yi = }>(•'.
Ex. Solve the problems of § 4 by the method of this section.
6. Invariants. — A function of the variables is said to be an inva
riant of a. group (or invariant under the group) if it is left unaltered
by every transformation of the group.
Thus, it is immediately obvious that any function of x alone is invariant under
the groups I and III, while any function of x'1 +/' is such under II.
We saw (§5) that
(7) A*i, *)-/(*, ;') - W+ wy^ + ....
In order that f(xlf )i)=/(-v, y) for all values of x and v, and the
corresponding values of xl and y\ into which they are transformed by
each of the transformations of the group, i.e. for every value of /, it
is necessary and sufficient that each coefficient in the right-hand
member of (7) be zero for all values of x and y. in particular, it is
necessary that
Moreover, since U~f=UUf, U"f=UU~f, •••, it follows at once
that (9) is also the sufficient condition that f(x\, y\)=f(xt y) for ail
values of x, y, and /. Hence, the
THKORKM. — The necessary and sufficient condition 1hatf(xty) be
invariant under flic group I / is L'f= o.
«
Remark. — This theorem may also be expressed as follows: 77/r
necessary and sufficient condition that f-.\\ \} he invariant under a
one-parameter group is tJiat it he left unaltered h\ the infinitesimal
transformation of the group. On succeeding pages will be found
conditions for invariance of curves, families of curves, differential
equations of various types, and so on. In each case it will be found
§§6,7 THEORY OF ONE-PARAMETER (IKOUl'S 17
(although specific mention of the fact will not be made) that the con
dition for invariance under the group always reduces to that of in-
variance under the infinitesimal transformation of the group.
To determine invariant functions, it is necessary to solve the partial
differential equation •) /• ^f
The corresponding system of ordinary differential equations is
(10) ^ = ^ = tf.
t 1J O
f= const, is one solution of the system.
If, besides, u(x, y)= const, is the solution of the equation involving
the first two members, the general solution- of (9) is, by Lagrange's
method,*
In I and III
o 77 o
.'. H = JC\ and/= l<\x).
In II u~x^ -f/2 ; and/= F(.
In IV « = ^; and/=
Ex. Find the invariants of the groups in the problems of § 4.,
7. Path-curves. Invariant Points and Curves. — As was seen in
$ 4, the differential equation of the path-curves of a group is readily
obtained from the infinitesimal transformation of the group. Thus,
u.iing .v and v as the variables, it is
dy _ rj
(ii) = .
t t]
* See El. Dif. Eq. \ 79.
1 8 THEORY OF DIFFERENTIAL EQUATIONS §7
The general solution of this equation,
u (x, y) = const.,
is the equation of the family of path-curves. As u (x,y) is an invariant
of the group (§ 6), it follows that the equation of a path-curve is
obtained b\ equating an invariant to a constant. Moreover, it is clear
that this property is characteristic of an invariant ; that is, if equa
ting a function to any constant whatever gives the equation of a
path-curve, that function must be an invariant.
But this is not the only form in which the equation of a path-
curve may appear.* A path-curve is an invariant curve of the
group, hence its equation must be invariant. \if(xt y) = o is to be
an invariant equation, f(x^ ji'j) must vanish for all values of x± and ft
into which the various values of a: and y which satisfy y"( jc, jj1) = o are
transformed by the transformations of the group. Now, we have seen
(?) /(-*•„ JO =/(*, y) +
If the right-hand member is to vanish whenever/^, y) does, for every
value of /, it is necessary and sufficient that each coefficient should do
so. *In particular, it is necessary that
(12) Uf= o, whenever f(xt y) = o,
that is, £^must contain f(xty) as a factor. |
But if Uf=<*(x,y)f(x,y),
then Wf = UUf= Uuf+ o) Uf= ( U<» + or )/;
i.e. £/yalso contains f(x,y] as a factor.
* Thus, while ^ = c is readily seen to he the equation of the family of path-curves
of the group Uf=x ~\~)' ' , y—cx — o is another form for it. U(y — cx)=. — ex -\-y
docs not vanish for all values of .i' and;1; but it does vanish for those values satisfying
the i-quation ot the path-curves ; sec; (i-j) bi-lo\v.
f It is presupposed that /(.v,y) contains no repeated (actors.
§7 TIIKORY OF ONE-PARAMETER r.ROUPS ig
In the same way it can be shown that every coefficient in (7) con
tains/^, y) as a factor, whenever 6fdoes ; for if
u*f= e(x, y)f(x, y), u**f= vv*/= ( m + 0o,)/.
Hence the vanishing of Uf whenever f(x, y) docs is both the nccessaiy
and sufficient condition thatf(x,y) = Q be an invariant equation.
In case Uf= o for all values for x and jy, the above condition is
fulfilled. But this we recognize as the condition (9) that f(xt y) be
an invariant. Hence, not only is f(x, y) = o a path-curve, but
f(x, y) = any constant^ one in this case.
Remark. — It should be noted that
may vanish because £ = o and r; = o * for certain values of the vari
ables. In general these two equations determine a finite number of
values of the variables. Remembering the significance of £ and rj,
these values of the variables are left unaltered by all the transforma
tions of the group ; so that the points having these values for coordi
nates are invariant points. If it happens that £ and rj contain a
common factor, <»(x,y)t it is obvious that u>(xty) = o is an invariant
curve, in that every point of it is invariant. Following Lie, and
desiring to preserve the significance of the name, we shall not include
this class of invariant curves among the path-curves.
Summing up the results of this and the preceding section we
have the
THEOREM. — The necessary and sufficient condition that /Y.v, v) = o
be invariant under the group Uf is that Uf= ofor ail values of x and
y for which f(x, y) = o, /'/ being presupposed that f(xt y) has no
repeated factors.
* Still another possibility is that J = o and *• = o \vhrnrvor f— o. But this is
9* dy
excluded by the restriction thut/"(.i-,_y) have no repeated factors.
2O THEORY OF DIFFERENTIAL EQUATIONS §§ 7, 8
Points whose coordinates satisfy the hvo equations £(x, y) = o,
77 (x, jv) = o are invariant under the group. If g(x, y) = o and
rj(x, y) = o whenever f(x, y] = o, ////> CV/^T^ /V composed of invariant
points. Curves of this type are not included among the path-curves
of tJie group.
In all other cases /(x, y) = o is a path- curve.
If Uf— v for all values of x and v, f(x, v) is an invariant, and
f(x, y) = any constant (including zero) is a path-curve.
In I, £EEO, rj= i.
.'. n~x = i-onst. is the equation of the path-curves.
There are no invariant points.
In II, £ = — y, f] = x.
:. n — x1 +jj'2 = const, is the equation of the path-curves.
There are no other invariant curves. The point x = o, y — o is invariant.
In III, £EEO, 77=^.
/. u~x = const, is the equation of the path-curves.
y = o is an invariant curve, each point of which is invariant.
Ex. Examine for invariant curves and points the groups appear
ing in the problems of § 4.
8. Invariant Family of Curves. — A family of curves is said to be
invariant under a group, if every transformation of the group trans
forms each curve into some curve of the family. We shall consider
at this time families containing a single infinity of curves only, that is,
those whose equations involve a single parameter or arbitrary constant.
Writing the equation of the family in the form
/(*, y) = t,
it will be invariant, if
/(*i, .>'i) =/!>(*, y> *\ «K*> y> 0] = <*>(•*> y, 0 = f>
is the equation of the same family of curves for every value of /, c
and c' being arbitrary constants.
A single infinity of curves determined by an equation involving
an arbitrary constant is equally determined by a unique differential
THEORY OF ONE-PARAMETER GROUPS
21
equation of the first order, of which the equation involving the arbi
trary constant is the general solution. lff(x,'y) = c and <o(jc,y, t)=c'
are to be the same family of curves, these equations must be solutions
of the same differential equation of the first order. Hence the left-
hand member of the one must be a function of that of the ether,* i.e.
w = F(f).
Making use of the relation (7) § 5, viz.
we see that/^, }\) will be a function of f(x, y) for all values of / if
and only if each coefficient in the expansion on the right is a function
of/(jc, _)'). In particular we must have
(13) Uf=F(f).
If (13) is true,
Tfl-f TTfTf TTJ?( /\ ~~_\*K) TTf ~*_\Jl J?( f\
Uj = UUJ = Ur(T )= — — Uj = — —S(j),
df df
which is again a function of/
In the same way each coefficient on the right is seen to be a func
tion of/; for if £/"/=<&(/), U"+lf= UUnf=
Hence (13) is both the necessary and sufficient condition that the
family of curves /•/ \
be invariant.
* The differential equations arising from these equations are
In order that these be one and the same equation it is necessary and sufficient that
d/
By
dw
dy
But this is the condition that w be a function of/. See El. />.>?'. /•.</., Note 1 of the
Appendix.
22 THEORY OF DIFFERENTIAL EQUATIONS §8
Rejnark. — A special case should be noted. If Uf = o for all values
of x and yt/(x, _)')— c is a family of path-curves, each one of which
is invariant, hence the family is. This particular family is charac
terized by the fact that its differential equation is
The problem of finding all the families of curves invariant under
a given group Uf will be considered later in another form (§ 18).
The general type of such families* may be found by noting that
f(x,y) must satisfy (13), where P(f) is some function of/, not de
termined. As a matter of fact, F(f) may be taken as any convenient
function of/, as may be seen from the following consideration :
The family of curves f(x, y)= c may equally well be written
<£[/(.*:, j)]= const., where <£(/) is any holomorphic function of/.
Applying (13)
«*(/)*= cy-=*C/).
This will be any desired function of/ say O(/), if
Since the family of path-curves is excluded, F(f)^. o. Hence the
function 4> can be obtained by a quadrature, such that when the equa
tion of the invariant family of curves is written 4>[/(,r, y]\ = const.
the right-hand member of (13) will assume the desired form O(/).
In the case of I, equation (13) is Uf= & = F(f}.
dy
From the corresponding system of ordinary differential equations
<tx_d£_ df
01 /?(/)
* In this discussion the family of path-curves is excluded, since a method for find
ing these curves has already been given ($ 7).
§§8,9 THEORY OF ONE-PARAMETER GROUPS 23
the general solution is seen to be of the form
where ^ is an arbitrary function, and <£ = f — •£— Solving for /, this takes the
J *\fj
form f=$(y — ^(x}}.
The most general family of curves invariant under the group Uf= =*• is then
dy
*(y — lK-*)) = const., or simply y — ^(x) = c.
Geometrically this is obvious at once. For such an equation represents a
family of curves all of which may be obtained by moving any one of them con
tinually, in either direction, parallel to the axis of y.
In II, — y^f- -\-x §f-= F(f} leads to — = ^-' = — ^— , whence the general
d-x dy —yx /'(/)
solution is of the form tan'1 ¥. — 0(/) =
or / « * (tan-1 £-
The equation ~v = t, representing the family of straight lines through the origin
x
is a simple example under this head, as is immediately obvious geometrically.
As an exercise, the student may show that
is a general type for III, while x^\ ~ ] — c
is such for IV. Simple examples are
A-- + 2- = I, a family of central conies of fixed transverse axis for III,
<t.\-~ + |8v2 = r, a. family of similar central conies for IV,
as is readily obvious geometrically, and as may be verified easily analytically.
9. Change of Variables. — The form of the transformations of a
group depends upon the choice of variables that are operated upon
by them.
Thus it is obvious that while the group of rotations II affecting the rectangu
lar coordinates is
.TI =r x cos a — y sin a, y\ = x sm a + y cos a,
24 THEORY OF DIFFERENTIAL EQUATIONS §9
when operating upon polar coordinates, it is
/>i =p, Oi = 0 + a,
which, in form, is identical with the group of translations I.
To find the effect of the change of variables *
(14) x=F(x,y), y=*(x,y)9
which, of course, carries with it
on the form of the finite transformations of the group
(i) xl = <j>(x, y, a), yL = \f/(x, y, a\
x, y, xl} \'i must be eliminated from the six relations, (14), (14'), (i)
and the resulting two relations solved for xl and y^ This elimination
is usually effected by solving (14) and (14') forx,y, xl} ylt and substitu
ting these in (i).
*The introduction of new variables in a transformation involves the following
processes :
Designating by S the transformation of variables (14), or (14'). a°d by 5""1 its inverse
x = F(x, y), y =
obtained by solving (14) for x and y, the new coordinates (x, y) of any point are ex
pressed by means of S~l in terms of the old coordinates (x, y). These in turn are
transformed by (i) or Ta ($ i) into (x1, y^) of the new point. Finally S transforms the
latter into (x\, y\), the new coordinates of this point. Designating by Ta the transfor
mation in the new variables corresponding to Ta in the old, the above may be expressed
symbolically
The transformation Ta is known as the transform of Ta by S.
That the aggregate of the transforms of all the transformations of the group (i) form
a group follows, of course, from the fact that the transformations imply certain opera
tions which are independent (except as to form, but not as to effect) of the kind of
variables operated upon by them. It is very easy to verify this, however, as follows:
TaTb = S-i /a.svf-i TbS = S-i 7 a TbS = S--1 TeS = Tc
since 6'A'-1 is the identical trunslormation, and 7'0 /i> — Te (§ i).
§9 THEORY OF ONE-PARAMETER GROUPS 25
In the case of the above example the formulae for the change of variables will
be chosen in the inverse form
x — p cos 6, }' = p sin 0.
Eliminating x,y, x\,y\,
Pi cos 0i = p cos 6 cos a — p sin 6 sin a = p cos (0 + #)»
pi sin 0i = p cos 6 sin # + p sin 0 cos a = p sin (0 -f- #).
Whence, solving for p\ and 0i,
pi = P, 0i = 0 + «.
(The other possible solution, pi = — p, 0i = 0 -f- IT +a, while exactly the same
geometrically is not to be used here, since the above transformation must reduce
to the identical one for a = o. In the above transformation of variables, it is
understood that p = + W2 -f-/2)-
M
In general, the actual work required to carry out this process is
long, to say the least ; on the other hand, the problem of finding the
new form of the infinitesimal transformation is a very simple one.
For, remembering that
a*
Similarly r\(x, y) = Uy.
Hence
,t
dx dy
where Ux and Uy are to be expressed in terms of x and y by means
of (14).
26 THEORY OK DIFFKRF.NTIAL EQUATIONS §§ 9, 10
In the above example, choose (14) in the form
p =
Since u=
10. Canonical Form and Variables. — It is always possible theo
retically, and often practically, to find the change of variables that
reduces the group to a desired form. Thus, in order to have the
group take the form
any convenient pair of independent solutions of
=^ y),
may be taken as the new variables x and y. In particular, to reduce
the group to one of translations in the direction of the axis of y,
when it takes the form U/= -<-, the equations to be integrated are
061)
The first of these is (9), § 6 ; so that for x may be taken any con-
vc'nient invariant of the group, //(x,y).
§ io THEORY OF ONE-PARAMETER GROUPS 27
To solve the second equation, I/igrange's method leads to the sys
tem of ordinary differential equations
dx __ dy _ dy
7 = 7~T'
which are equations (5), § 4. Making use of the fact that u (x, y) =
const, is the solution of — = ^~, y may be obtained by a quadrature.*
£ *7
Following Lie we shall say that the group is in the canonical form
when it has the form Uf= — , and the variables which reduce it to
oy
this form will be called canonical variables. The above result may
then be stated :
Every group can be reduced to the canonical form Uf^ — * In
g dy
order to Jin d the canonical variables, it is only necessary to solve the
differential equation of tlie first order
dx _ dy
*"*'
and to follow this witJi a quadrature. In case an invariant of the
group (or what is the same thing, the equation of its path-curves) is
known, a quadrature alone is necessary.
Remark. — If the equations (16) cannot be solved readily, it may
be practicable to find the canonical variables for both the original
and the desired forms of the group. A proper combination of these
will then give the required transformation of variables.
In II, £= — y, TJEE.T. Here, as was seen (§ 4), WEE.T- + J'-. v tan""1*^-
x
These ate a possible set of canonical variables. I hit it is customary to choose
V// instead of u for X, thus giving the usual polar coordinates. In III, £^O,
* Inspection of equations (6), ^ 4 slioxvs that the transformation X "(•*•', y),y =
v(x,y) reduce.-, tiic i;roup to lli-: torin
I xi - x, j/i = y 4- 1.
ft
28 THEORY OF DIFFERENTIAL EQUATIONS §§ 10, n
77 = y. Here, as was also seen (§ 4), u = x, v = \ogy. In IV, — — -^- gives
x y
u=-*', which may be taken as x. By composition the system of equations
dx dy dy . dx -f dy du ,
— = -*- — — gives - £— £ = -3tj whence y = log (x +}').
x y I x -j- y I
Another set of canonical variables for this group is of some interest. By com
position, after having multiplied numerator and denominator of the first member
by x and of the second member by y, we have x x ^ ? — — ; whence y =
x-+* I
log Vy2 + y'1. Choosing this form for y and tan"1 u = tan"1^ for jr, the canoni-
x
cal variables are very similar to the usual polar coordinates, in that the old
variables, in terms of them, are
x = eU cos x, y — eV sin x.
From their nature, it is obvious that in passing to the usual polar coordinates
the transformations IV assume the form of the affine transformations^II, as may
also be verified readily analytically.
Ex. Find the canonical variables of the groups in the problems
of § 4-
11. Groups Involving More than Two Variables. — The previous
theory of one-parameter groups involving two variables can be gen
eralized in two directions : the number of variables can be enlarged,
and the number of parameters can be increased. In this section*
will be considered one-parameter groups involving more than two
variables ; and as the argument is almost the same for n variables as for
three, the latter number will usually be employed. As a matter of fact,
the previous arguments for two variables hold, with only slight modi
fication, for a larger number ; hence, as a rule, only the facts will be
given here, it being left as a reviewing exercise for the student to
fill in the supplementary arguments.
* A brief extension of the above theory to groups involving more than one-pa
rameter will be given in Note VI of the Appendix.
§ii THEORY OF ONE-PARAMETER GROUPS 29
Starting with the transformations
('#! = <£(.*, y, z, a),
yi=*$(x,y,*> <*),
z\ = \(x>}'> z> a\
where </>, ^, x are supposed to be generally analytic, independent,
real functions of x, y, z, a, they will constitute a Lie group provided
the set has the following properties :
i° The result of carrying out in succession two transformations of
the aggregate, determined by any two values a and b of the parameter,
is the same as performing a single transformation of the set determined
by some value c of the parameter, where c is a function of a and b.
2° Solving [i] for x, y, z in terms of xly ylt slt the resulting
formulae take exactly the same forms as [i], some function of a tak
ing the pfece of a. In other words, the transformations of the group
occur in pairs of mutually inverse ones.
As a consequence the group contains the identical transformation.
A group of this type contains one and only one infinitesimal trans
formation (§ 2, and Remark, Note I of the Appendix), which may be
written *
[3] Uf= t(x, y, ») + , (,, y, *)+{(*, y, s)^,
where, in general,
da \da
\ »
ri"da=(daja:
da \da
* For « variables we have likewise
TIIKOKY OF DIFFERENTIAL EQUATIONS
The finite transformations of the group may be obtained from the
infinitesimal transformation either in the form of a power series in the
parameter (§ 5) /2
xl=*x+ Uxf + U*x — + •••,
2 !
*! ^ 2 [
or as solutions of the differential equations (§ 4)
r n dx^ d\\ <h, _dt
-" £( i \ — ~~7~~- ~> \ — V(~~i) '•"S — — *
If ii\_(xi, y\, 2j) = const, and /v^v,, rh z{) = const, are the solutions of
the first two equations (not involving /), and r(.\'lt}\t -/) — /= const.
is a third solution of the system independent of the other two, then
determine the finite transformations of the group.
In both these cases the parameter / enters in such a way that /= o
gives the identical transformation, and /= — /determines the inverse
transformation.*
* In the case of n variables, the development form of the finite transformations is
exactly the same. To obtain the second form, the system of differential equations is
[Si
and their solutions are of the form
V
^(.iV, .v./, ..-, v,/) = Ul(.rlt jro, ••-, xn),
Primed letters are used here to designate the transformed variables, since the sub
script, previously employed, is no longer available.
§n THEORY OF ONE-PARAMETER (iKOLTS 31
The effect of a finite transformation of the group on any function
/(x,y, z), is(§ 5)
[7] /(*Wi, sj =/(*,>', *) + Uft+ U*f~± "-
A function /(*, y, s) is invariant under the group Uf\i
for all values of .r, v, s (§ 6).
This equation, involving three independent variables, has two inde
pendent solutions. Hence a one-parameter group in three variables
has two independent invariants. Since MI(X, y, z) and //o(.v, v, z) are
such a set, every invariant of the group is a function of //i and //2.*
Those points whose coordinates satisfy the three equations
£(.v, y, z) = o, rj(x, y, z) = o, £(x, y, z) = o
are invariant under the group (§ 7). In general, that is, in case the
three functions are independent, there is only a finite number of
such points. But if only two of the functions are independent (which
will show itself by having their Jacobian vanish, without all of its first
minors doing so) the two independent equations will be the equations
of a curve, every point of which is invariant. If all the two-rowed
determinants in the Jacobian vanish, there is only one independent
equation, and it is the equation of a surface, every point of which is
invariant under the group.
The path-curves are obtained
i° either by eliminating a from the finite transformations of the
group (§ i),
2° or by solving the system of ordinary equations (§ 7)
-i '/v t/ dz
* In the c«M <>f // variables, every invariant of the group is a function of the »— i
independent ones //,, //.>, •••, un j.
32
THEORY OF DIFFERENTIAL EQUATIONS
§u
From the latter we see that if //j and //.2 are two independent invari
ants of the group, //j = const, and u2 = const, are the equations of the
path-curves.
Each of the surfaces u\ — const, and /A, = const, is invariant, being
made up of an infinity of path-curves obtained in either case by keep
ing one of the constants in the equations of the path-curves fixed and
allowing the other to run through its full range of values.*
The equation f(xt y, z) = o, or the surface represented by it is
invariant (§ 7) if
[12] Uf— o whenever/= o,|
provided / contains no repeated factors. (If Uf vanishes because
£ = o, 11=0, £=o whenever /=o, every point of the surface is
invariant.)
The curve/j (x, y, z) = o,/ (x, y, z) = o is invariant if
[12'] Ufa = o and £/2 = o whenever/ = o and/2 = o,
provided/ and/, contain no repeated factors and are independent
functions, not containing a common factor. This last condition
assures us that not all of the two-rowed determinants in the matrix
dx
df,
dx
By
df2
dy
dz
df,
dz
vanish for all values of x, y, z.
* In the case of « variables, i° holds without change ; in 2° the differential equations
of the path-curves are
£
£•
and their finite equations are «j —const., //._, • < (v/v/., ••• , //„ _i const., where //]//.,>, ••• ,
n,, i. are any»— i indej>endent invariants. l-!acli <>i t!u; (// i) -way spreads in n
dinii-nsioMs //! = const., u^ — consf., ••• , ««-i = const, is invariant, as well as the various
spreads of lower dimensions obtained by taking these invariant relations two, three,
• . // -i together, the last case giving the path-curves.
t This condition holds when the equation involves any number of variables.
§u THEORY OF ONE-PARAMETER ClROUPS 33
The argument employed in establishing this theorem for a curve in
three dimensions is different from that available in the case of a
surface f(x, v, z) = o (in the latter case the one employed for a
curve in two dimensions (§ 7) applies).
The necessity of the condition is seen as before ; for, using
formula [7]
H f\(x\i )\i z\} and fi(x\,y\, Zi) are to vanish whenever /(jc, j, 2)
and /*(x,y, z) do, for all values of /, it is necessary that f/l/=o
and £/2/= ° whenever/! = o and/, = o.
The sufficiency of the condition follows at once from the fact that
since *, , -,
all along the curve /1 = o,/ = o, £, 77, ^* are proportional to the
direction cosines of the tangent of this curve at each point (x, y, z) ;
that is, this curve is the path-curve through the point (x, y, z).
Remark. — If Vf\ = o whenever ^=o, and Uf« = o whenever
/ = o, the surfaces / = o and / = o are separately invariant ; and
their intersection is also invariant. In the case under consideration
above, however, [12'] is the condition for invariance of the curve
without regard to the nature of these surfaces.
The change of variables
[14] x = F( >•, y, 2), y = ^.v, v, z\ z = vl/(.v, r, s)
* If | — o, 17 = o, f = o whenever/! = c, /^ = o, every point on this curve is invari
ant, and hence, the curve itself is ; so that the sufficiency is also established in this case.
But such a curve is not included among the path-curves of the group (Remark, § 7).
34 THEORY OF DIFFERENTIAL EQUATIONS § n
•
causes the infinitesimal transformation to take the form (§ 9)
[IS] ^ff+^I^S-
So that the new variables satisfy the differential equations (§ 10)
& = &+* + & = &,,,:),
ox oy oz
In particular, when | = o, i] = o>£=il the group is said to be in
the canonical form* If the equations of the path-curves are known,
the canonical variables can be found by means of a single quadrature.
To illustrate all that has gone before consider the group of screw motions
x\ — x cos t — y sin /,
y\ — x sin / + y cos /,
21 = 2 + ;///,
where m is any constant.
The student will have no difficulty in proving that these transformations have
the group property, and that in this case (§ i)
Ttjtt= Ttl+tt',
also l— — t, and /0 = o.
The infinitesimal transformation is readily seen to be
*Morc generally, the group will !•<• said to !x; in tin- canonical form when any one of
£> 77» 4" <'<J'«;ils ;i roustunt, and the other two are zero.
§n THEORY OK ONE-PARAMETER r.uorps 35
Conversely, starting with the infinitesimal transformation the Unite transforma
tions are found to be, using [<SJ,
( t"1 t* \ I /3 /5 \
^xil --- (- — — • • • — y\t --- 1 --- "• 1 = * cos t — y si
\ 2! 4! J \ 3! 5! j
(t* /.">
/ -- -+ :—•
sin t.
/2 /4
_. . , , • - ,
2! 4!
z\ •=• z -\- nit = z -\- nit ;
r, using the other method,
= tan-1 Pi- — ^L = tan -1 -' — —,
,V| m x m
For practical purposes it will he simpler to replace z, in the second equation
hy its value in the third one. Then
tan-1^ = tan"1-'1' + t,
x\ x
Zi = Z + ////.
The third equation is already in proper form.
Ilr- lusi t\vo equations are free of c, and, as was found in § 4, reduce to
XL = .v (-us / - - )' sin /,
jj'! — x sin t -\- y ci >s /,
T\vo iiKh'pendent invariants are n\ .\- '• i':, // . 'tan l!/ —• Hence th
, x in
pain-curves are
*2+;(2 = r2f ^-l^^i.^,
X III
t—, introducing the parameter 9,
x — rcos/'. r = rsin^, z = »i(0 — r\
lich is a family of helices, involving the arbitrary constants r mid r,
If m ^ o there are no invariant points.
36 THEORY OF DIFFERENTIAL EQUATIONS §11
Two of the canonical variables, x and y, must satisfy the differential equation
_ ,df + xQf±.,n Of- o
d* dy dz
while the third, z, must satisfy
df , df df
— V — -f- x •— 4- m — = i.
d* dy dz
Knowing the invariants of the group, u\ and u», we may put
x — VV2 + y2, z/ = tan'1 % — — •
x m
By inspection, z may take the simple form
Solving for the old variables, the formulae of transformation of variable0 »*e
x = xcos (y + z), y = ysin (y + z), z = mz.
It is obvious that the change to cylindrical coordinates
x = p cos 6, y = p sin 0, z = z
reduces the group to the form
Pi — Pi 6\ •= 0 -\- t, z\ = z + w/"»
which is a group of translations, but not in the canonical form.
Discuss as was done in the text the following groups :
"C* Y O y* /7 Y* U — il V ? --, i*
Ex. 3. xl = rt.v, T! = ay, zl = az.
Ex. 5. xl = ea (x cos a —y sin «), yl = e° (x sin a +y cos «), ^ = eaz.
CHAPTER II
DIFFERENTIAL EQUATIONS OF THE FIRST ORDER
12. Integrating Factor. — We have seen (§ 8) that if <j>(x,y)
const, is a family of curves invariant under the group
(13) U*
Moreover, it was also shown in § 8 that if the curves of the family
are not path-curves of the group, the equation of the family can be
chosen in such form that the right-hand member of (13) shall be
come any desired function of (f>. In particular, there is no loss
in assuming the equation so chosen that this right-hand member is
i • for if a given choice <£ = const, leads to F(<$>\ the selection
<$(<£) = const., where <$(<£) = I l *** , will give £/<£>(<£) = i.
J F(<f>)
Suppose now that
(17) Mdx -\-Ndy-Q
is a differential equation whose family of integral curves
( 1 8) $(x , y) = const.
is invariant under the group Uf, the integral curves not being path-
curves of the latter. Let <£ be so chosen that
(19) ^ = ««*+,|4=I.
dx By
37
38 THEORY OF DIFFERENTIAL EQUATIONS § 12
Since (18) is the solution of (17),
,. d<f> , . dd> ,
d<\> = -^-dx + --?-dy = Q
ox ay
must be the same equation as (17) ; hence
0<j> d$
dx dy
(20)
-
dx dy
From equations (19) and (20) the values of -— and — are found
ox dy
to be
Mdx
Hence the
THEOREM.* — If the family of integral curves of the differential equa
tion Mdx + Ndy = o is left unaltered />v the youp Uf~^- + »4^i
Av dy
is an integrating factor of the differential equation.
Remark i. — This theorem ceases to hold in case the curves (18)
are path-curves of the group Uf. In this case (19) becomes
£—+77 — = o; whence, taking account of (20), £Af+-nN=o.
dx dy
As a matter of fact, it is obvious that in this case the curves (18),
being the integral curves of (17), are the path-curves for every group
of the type
* This theorem of Lie was first published l>y him in the Vcrhandlungen der Gi-sell-
scliaft dcr XN'isscnhchaftun zu Christiania, November, 1874.
§12 DIFFERENTIAL EQUATIONS OF THE FIRST ORDER 39
where p(x, y) is any holomorphic function of x and y. Such groups
are said to be trivial for purposes of assisting in solving the differen
tial equation (17).
Remark 2. — At times it is obvious from the nature of the problem
that the family of integral curves is invariant under a certain group.
This will be found to be the case in the following examples :
Ex. 1. Find the curves whose tangent at each point makes an
isosceles triangle with the axis of x and the radius vector to the point
of contact.
This family of curves is clearly invariant under the similitudinous group
£//=.*•— -f yQ£- Its differential equation is
dx * dy
dx x dy I dv \ 2 , dy
! +Z£L~~ </x' \<tx) ' <t*~
x dx
Reducing to the form (17), which is characterized by being of the first degree
in dx and dy,
(x ± V** + jP) dx + y dy — o.
The integrating factor
I I
+
Integrating, log (x± V-r'-^+y2) = const, or x ± V-rM
This reduces at once to v2 = c- — 2 r.r, a family of parabolas having the origin
as common focus and the axis of .r as common axis.
Ex. 2. Find the curves such that the radius vector to each point
makes an isosceles triangle with the tangent at the point and the
axis of x.
4<D THEORY OF DIFFERENTIAL EQUATIONS §§ 12, 13
Ex. 3. Find the curves such that the length of the radius vector
to each point equals the tangent of the angle between the radius
vector and the tangent to the curve at that point.
Ex. 4. Find the curves such that the radius vector to each point
makes a constant angle with the tangent to the curve at that point.
Ex. 5. Find the curves such that the perpendicular distance
from the origin to the tangent to a curve at any point is equal to
the abscissa of that point.
13. Differential Equation Invariant under Extended Group. —
While at times it is possible to tell from the nature of the problem
whether the integral curves of a differential equation form an invariant
family under a certain group, it is desirable in order to extend the
usefulness of the theorem of the previous section, to be able to tell
when this is the case from the form of the differential equation itself.
A point transformation
carries with it the transformation
cty
d\, dx dv
dx dy
tt+tty
. dx rV .,
*"-§73£"3^#-
dx dy
* This is called a point transformation because it transforms the point (x,y) into
( r,. r, ). It thus transforms the various points of a curve F(x,y) = o into the corre
sponding points of some other curve /'\(-*'i,>'i) = o, and may therefore be said to trans
form the curve F(x,y) = o into ^(x^y^ — o.
§ 13 DIFFERENTIAL EQUATIONS OF THE FIRST ORDER 41
where v' = — and r/ = — • Since ^ is a function of x, y, v' only, it
dx dx^
follows that the point transformation implies the transformation
fy , fy
~
, vt
"•T" i — a"~«'
djc ay
affecting the three variables x, y, y'. The latter transformation is
known as an extended point transformation*
Starting with the one-parameter group of point transformations
(i) .T! = 4>(x, y, a), )\ = $(x,y, a)
it is easily seen that the corresponding extended transformations
(21) xv= <j>(x,y, a),yi = t(x,y, *)iJfc'- = ^-3jit(#,j».7/«)
also constitute a one-parameter group in the three variables xt y, y'.
For, since the equations of a point transformation are precisely the
first two of the corresponding extended transformation, and since the
third equation of the latter is determined uniquely by the first two,
the fact that the transformations (i) have the group property (§ i)
predicates the existence of the group property in the case of (21).
Thus if a and /; are any two selected values of the parameter, the result of per
forming successively the two point transformations
*i = 0O, y, «), y\ — <AO, ;', «)
and x« = 0(>i, _>'!, /;), y» = \f/(xi, yi, b)
is *2 = «/>(-*-, y, Ot yt = $(*> y> 0
* An extended point transformation is a special kind of a contact transformation
($ 49) ; for it transforms (.v, v, /) into (.»•],;',, ;•,'), where, if (.v,j) is some point on some
curve /''(-ft y) = o, y' is tin- .slopr of tlir t. indent to tin- curve at th.it point and 1^' is the
slope of curve /^(.Vj, y\) = o (into which the other is transformed by thr point trans
formation) at the corresponding point (-»"], /j). Since the value of _V]' depends upon
x,y,y' only, any curve tangent to F(x, v) =oat (.v,y) will be transformed into a curve
tangent to f>\ (-«'i,>'i) = o at the point (.Vj, _vj).
42 THEORY OF DIFFERENTIAL EQUATIONS § 13
where c is a function of a and b. This follows from the group property of (i).
In the case of the corresponding extended transformations
. ,t
"0 (x,y, a
and x.
the result of replacing .TI and j'i in the first two equations of the second trans
formation by the values given in the first transformation is therefore
Hence in the last equation of the second transformation,
In exactly the same way, the fact that a value of the parameter
exists giving the identical transformation for the group (i), and also
the fact that the transformations of (i) can be separated into pairs
of mutually inverse transformations, assure these same properties for
the transformations of (21). The latter therefore constitute a Lie
one-parameter group. This group is known as the once-extended
group corresponding to ( i ) .
With Lie, we shall write as the symbol of the infinitesimal trans
formation of the once-extended group
(22)
where, as before, t = ^, ^= -, while r,' = ^ =&,(
It was seen in § 4 that, with a proper selection of the parameter,
, = ( , ami, for any function/, =
§13 DIM'KRKNTIAI. F.OUAT1OXS OF Till'; M RST ORDER 43
In a sense then 8 is a differential operator, so that 8 and d are
commutative operatois ; thus, for example,
, 8 dy\ 8<r ' - &*v V
Hence r? = ^ * , = — -, 7-7 — -^ — — — j , — -:
' badx tfa (ifcr dx dx dx
dx J dx
Remark, — Attention should be called to the fact that, while / is
ual to -^, -n is usually diff
dx
hand member of (23), we have
equal to -^, -n is usually different from -£• Expanding the right
dx dx
where it is to be noted that rf is a quadratic polynomial in 7' when
Given a differential equation of the first order
(25) /(*,*/>*»o,
the effect of any transformation (i) on the variables .v and v is to
transform the differential efjuation (considered as an equation in the
three variables .v, r, / ) by the corresponding extended transformation
(21). The family of integral curves of (25) is invariant under the
group if each integral curve? is transformed into some curve of the
family by every transformation (i). Hence every transformation (21)
44 THEORY OF DIFFERENTIAL EQUATIONS §§ 13, 14
must leave the differential equation unaltered. The condition for
this is ([12], § n)
(26) ir/~ £ + j + rj' = o whenever/(>, jr,y)= o.
Hence the
THEOREM. — The family of integral cuives of the differential equa
tion f(x, y, y') = o, and, therefore, the differential equation itself, is
invariant under the group Uf if U'f= o whenever f= o.
In the case of II, £=— y, TJ = X. Hence, from (23) r?' = i +?'"' The ex
tended group of rotations is then
U^.SfS^df
' d* dy d/
The differential equation of the family of lines <- = c (which is invariant under
II) is xy' — y = o. Here
Vl(xyi-y) = -yy» - * + (l +/*)* =/(^/-j).
This vanishes whenever xy' — y does.
14. Alternant. — Let U\ and 60 be any two homogeneous linear
partial differential operators *
^i = li(^, .)')— 4- »7i(.v, y)— ,
C/2 = &(#, >')— + ^(.v, 7)— .
Then
(27) ••• i^J
* Yor the sake of simplicity \v<> shall suppose that two variables are involved. But
tiiis entire section hoids without any inodilication tor -v variables.
§§I4, 15 DIFFERENTIAL EQUATIONS OF THE FIRST ORDER 45
Writing l\U,f- UMJ=(yjJ?)f*
the operator (U\U^), which is known as the alter nan t\ of i/\ and £/,,
is seen to he one of the same type as U{ and U>.
The following properties of alternants are immediate :
i, U,± tf8) =
] 5. Another Criterion for Invariance of a Differential Equation
under a Group. — A second form for expressing the condition that a
group leave a differential equation unaltered plays a very important
role in the further development of the theory. It was seen (§ 1 2),
that if
( 1 8) <£ (x, _y) = const.
is the solution of
<£ is a solution of the partial differential equation (20)
,7dcf> jij-dtb
( 2 o ) A<p = IV — — — J.V1 — - = O.
dx By
Moreover, if the family of curves (i 8) is invariant under the group Uf
(without being path-curves of the latter), <£ may be so chosen that
Consider now the alternant of f/and A (§ 14)
Af Af
(37)
* Lie writes ( U± U.,} or ( I \f, U.,J} instead of ( l\ ('.,) f.
f Also sometimes called the commutator of f \ and / j.
46 THKORY OK DIKKKRKNTIAL EQUATIONS §15
Because of (28) and (19) (UA)$ = U(o)—A(i)=o.
(29) A (UN-A^-WM+A^*^.
Since cf> is a function of at least one of the variables x and r,
-£ and — are not both identically zero. Hence the coefficients of
dx dy
(29) must be proportional to those of (28); i.e.
(3o)
or UN-A£ = *N, UM+A-n
Putting these in (27)
(31) (UA)f=\(x,y)Af..
Hence (31) is a necessary condition that the integral curves of (17)
be invariant under Uf.
Conversely, if (31) holds
because of (28). Hence AU<f> = o.
Since every solution of (28) is a function of <£
This is the condition [§ 8, (13)] that the family (18) be invariant
under the group Uf. Hence the
THEOREM. — The necessary and sufficient condition that the differen
tial ('({nation M dx + N dy = o be invariant under the group Uf is
(31) (UA)f=\(X,y)Af
Af=N^ -M-£.
dx dy
* The common ratio \(x,y) is, at most, a function of the variables. It may be a
constant or zero.
§ 15 DIFFERENTIAL EQUATIONS OF THE FIRST ORDER 47
The condition (31) was found independently of what has gone before. It
may be obtained at once by means of (26). It is suggested as an exercise, that
the student do this. Here f(x, y, /) E= Af+ Ny1 . The expanded form of ij',
given by (24), must be employed.
This theorem leads to another one, of some interest, which is, as
a matter of fact, the converse of the theorem of § 12.
If £(JT, j) and t] (x, y) are any two functions such that
i
is an integrating factor of
(17) M dx -}- N (fy = o,
d( N \ d( M \
or
dx dx dx dx dy dy dy
dy
Dividing by J/7Vand rearranging the terms,
dx dy dx dy __ dx dy dx dy
" N M
UN-Ai UM+Ai)
(30) -JT ~M~
from which follows (31) as before. Hence, if /JL(X, v) is an integrat
ing factor of tJic differential equation M dx + N dy = o, and ±(x,y)
and rj(x, y) are any holonwrphic functions of tlic variables satisfying
the relation
* See EL Dif. Eq. \ 7.
48 THEORY OF DIFFERENTIAL EQUATIONS §§ 15, 16
the differential equation is invariant under the group
w£ + ,3£.
dx dy
Since £ and 77 are subject to the single condition (32), one of them may
be chosen at pleasure, and then the other is determined uniquely.
Hence, starting with an integrating factor of a differentia! equation
of the first order, an infinite number of groups can be found which
leave the differential equation unaltered.
It will be seen in § 17 that the general expression for such groups involves
two arbitrary functions. As a matter of fact, this can also be seen from the form
of (32). For if /j. is an integrating factor giving f»(Aft/jr + Ndy} = du, then for
/•'(#) any function of it, /"/•'(«) is also an integrating factor. (See El. Dif. Eq.
§ 5.) Using this as the right-hand member of (32), and selecting £(>, jj/) arbi
trarily, 77 = — ^ '/^ . The general type of group leaving (17)
unaltered may, therefore, be put in the form
T7J- >- f .. ..\ Of , I I
where £ and /'are arbitrary functions.
16. Two Integrating Factors. — Since the knowledge of a group
which leaves a differential equation unaltered gives an integrating
factor, thus reducing the problem of solving the differential equation
to a mere quadrature, it should be expected that the knowledge of a
second group which leads to a distinct integrating factor still further
simplifies the problem of solving the equation. This is actually the
case.
Suppose /M! and /xo to be two integrating factors of (17). Then
dy r).v
dy dx Ml ( 'dx ' " dy) /xA" ' dx " dy
§§ i6, 17 DIFFERENTIAL EQUATIONS OF THE FIRST ORDER 49
i dii d(log u) i 3u 3(log u)
Remembering that --C = -3 pLC/, -g — v fe^? and
|u, o.r • ojc /x qj oy
log /xx — log /x2 = log ^, the last equation becomes
/x,
AT 3' /, MI\ 1^-3 /, uA
3jcV )"" F V ) = ° '
/>., log — is an integral of
(28) A/ =
-
dx dy
Hence ^ is also an integral of (28), and
^2
^ = const.
P-2
is a solution of (17). So that the knowledge of two integrating factors
gives the solution of the differential equation without any analytic
work whatever.
Remark. — It is interesting to note that in the proof usually given
for the theorem that when one integrating factor /a is known, an in
finite number of others can be found [viz. if p.(Mdx-\- Ndy)=:duy
then pF(u) is an integrating factor where F(u) is any function of //],*
all possible integrating factors are found.
17. General Expression for Group under which a Differential Equa
tion is Invariant. — We have just seen that if U^f and £/,/are any
two groups which leave the equation (17) unaltered,
P.,
* See El. Dif. Eq. \ 5.
50 THEORY OF DIFFERENTIAL EQUATIONS §§ 17, 18
is a solution of (17) ; hence,
(33)
where <£(.Y, y) = const, is any selected form of solution of (17). Re
arranging the terms in (33),
&
where p(x, y) is the common value of the two fractions. Whence
(35) •
Conversely, if U^f leaves the differential equation unaltered, U.,f
given by (35) will also do so, no matter how /p(^>) and p(x, y) may
be chosen (it being understood throughout that all functions involved
are to be generally analytic). For, by hypothesis, using (31)
then (U,A}f = (F($}U» A)f+(pA, A)f
J+ P(AA)f-
Hence every gn>i/f> which leaves the differential equation unaltered
is given by (35), U\f being one group of tins sort.
If F($] is a constant, the resulting group gives the same integrat
ing factor as U\f.
If F(<f>) is identically zero, the resulting group is trivial (§ 12).
18. Differential Equations Invariant under a Given Group. — In
order to make use of the theorem of § 12, a group leaving the differ
ential equation unaltered must be known. While such groups always
§ iS DIKKKKKNTIAI. F.< H'ATIOXS OK TIIK KIKST ORDKK 51
exist, and are sometimes suggested by the nature of the problem
giving rise to the differential equation, the number of equations for
which they are known is comparatively small. The converse prob
lem of finding the general type of the differential equations invariant
under a given group is much more direct. And while its complete
solution requires the knowledge of the path-curves of the group and
usually one or several quadratures, it is practicable to supply these
in a large number of cases of interest.
It is clear that the differential equation obtained by equating an
invariant of the extended group (§ 13) to an arbitrary constant is
invariant. The general type of invariant of the extended group is
obtained by taking an arbitrary function of two independent solu
tions of ([9], § n)
Passing to the corresponding system of ordinary differential
equations
dx d dl
t(x,y) !,(*,?) V(
the first equation is recognized as (n), § 7. Its solution is
A second solution, independent of this one, must involve /. Writ
ing this in the form
U'(x,y,/)** COnst*,
the general solution of (36) will be of the form/(//, //'). Equating
this to an arbitrary constant gives the general type of invariant dif
ferential equation. There is ho loss of generality in equating/(//, //')
* Since u'(x,yty') is an invariant of the extended group / "/ and involves;'', it is
known as a ///-.*•/ Jiffo cntial invariant of the yroup I Y.
52 THEORY OF DIFFERENTIAL EQUATIONS §§18,19
to zero, the arbitrary constant being incorporated in the arbitrary
function/. So that the general type bf invariant equation is
(38) /(*,i
Several methods for rinding a' suggest themselves :
(a) Solving u(x, y} = c forj, and replacing it by its value in terms of x and
c wherever it occurs in the first and last members of (37), the Riccati equation
(39) = +_
d* td* t\dy fe By
results. In Note II of the Appendix it is shown that this equation can be solved
by quadratures.
(Z») The introduction of canonical variables (which can be found by a quad
rature when u is known, § 10) reduces the invariant differential equation to the
simple form
dy __ d* dy _
~~ ( }'
as will be shown, I, § 19. Since the one canonical variable x is the invariant //
or a function of it (§ 10), F(x} is a function of u. Because of the general type
of invariant differential equation (38), - - 2, — may be taken as u1.
dx+fc v'
d* dy* %
(c} Frequently some special method (see El. Dif. Eq. § 65) may be found
which is more direct.
19. Illustrations and Applications.
I. Uf= . . £ = o, y= i. .-. rj' = o. Equations (37) are
dx _ dy __ dy'
o ~~ i "" o
.-. // = x, ?/' =y. Hence the general type of differential equation
invariant muter Uj r= v- tsf(x, y') = o, or y* = F(x).
§I9 DIFFERENTIAL EQUATIONS OF THE FIRST ORDER 53
This equation is characterized by the absence of y. The variables
are separated when the equation is solved for /.
I'. 6^= -. It is readily seen* that the general type of differ
ential equation invariant under this group (of translations in the
direction of the axis of x) is y' = F(y).
This equation is characterized by the absence of x. The variables
are separable, thus -— — — dx.
II. ^=_JK|?+*J£. £=-X, rj=X. /.V=l4-/2.
'
dx dy dy'
Equations (37) are
-y x i+/2
/. u = jc+y2. To find //, multiply numerator and denominator of
the first member by — y, and those of the second member by x ; then
by composition (Et. Dif. Eg. § 65, 3°),
x dy —y dx dy1
r+72'
.*. «' = tan"1- — tan~Ty. It is simpler to take the tangent of this
"v _ - •'
function as the second invariant; i.e. «'=-^- -- -—. Hence the gen-
x+tf d, ^ d,
eral type of differentia! equation invariant under Uf= — y -r — ^XJT
* It is suggested as an exercise that the student actually carry out the work here
and in the cases below, where results alone are given.
Of course, the differential equation invariant under a group whose number is
primed may be obtained from that invariant under the corresponding unprimcd one
by interchanging x and r and v' and ' . I'.ut as an attempt is being made here to
make a collection of differential equations invariant under known groups, the foims
by which these differential equations are most readily recognized are given.
54 THEORY OF DIFFERENTIAL EQUATIONS § 19
Note. — This form of the invariant differential equation is obvious
from geometrical considerations, since // is the square of the radius
vector to any point on an integral curve, and //' is the tangent of the
angle between the radius vector and the tangent to the curve. Since
any function of// and //', containing v', can be used as a first differ
ential invariant, \/— — - or — ^ is available. So also is
Vi +/2
These are respectively the distance of the
normal and that of the tangent from the origin, each of which is left
unaltered by the group of rotations about the origin. Hence the gen
eral type of differential equation invariant under this group may also
be written
III. Uf=y-^-. £ = o, 77 =y. .'. T/ =y. Equations (37) are
<& <ty_
o ~~ y
Hence the general type of differential equation
invariant under irf~y^ W/f*i ~j, or }~ = F(x).
This equation is characterized by being homogeneous in y and y\
The variables are separated when the equation is solved for -^ .
*•. f J
III'. Uf=. Xjf-* It is readily seen that the general type of differ
ential equation invariant under this group (of affine transformations)
is xy' = F(y). . ^
The variables are separable, thus _ - = — .
1 /M .r) x
IV. Ufi~ .x -! +)?{-. Here V = o, and » =-r, //' = v'. Hence
av By A-
§I9 DIFFERENTIAL EQUATIONS OF THE FIRST ORDER 55
the general type of differential equation invariant under
This equation is characterized by being homogeneous in x and y.
Note. — An equation M dx + N dy — o is of this type when M and
TV are homogeneous functions of x and y, and of the same degree.
In this case the integrating factor of § 12 is — — — • (Compare
•-. Herei/=E — 2 /, and u = xyt //' = A~y.
Hence the general type of differential equation invariant under
t\f f) f *
Uf^ x : — v ; is f(x\\ .vV ) =o, or AT' = r/^ (AJ).
This equation is characterized by being homogeneous in A*, }', y\
when these elements are given the weights i, — i, — 2 respectively.
(Compare VI below.)
Note. — An equation M dx -f- N dy = o is of this type when J/=
(AT), ^=A;/!,(AT). In this case the integrating factor of § i 2 is
(Compare Rl. Dif. Eq. § 17.)
VI. Uf^x + nv -* f = A-, w= ;/r. .'. «'=(// - Or', and
c'Av " fJr
// = ••-, w's-^—j- Hence //(/• general type of differential equation
x x
* n may be any number. In particular // — i gives IV, while >t i ^ivcs V, and
n — o gives III'.
If the group be written in the more symmetrical form I r/~a.v Q- -\- Iry • t}1R
c).»- (>'
invariant differential «quation takes the fonn.vr' r /•'(•'' )• a- /• gives IV, a= - b
\ \-h/
gives \r, ,/ o gives 1 1 I, /> o gives 1 II '.
56 THEORY OF DIFFERENTIAL EQUATIONS § 19
invariant under Uf=x-£--\-nv-- /V/( — , —— ] = o.
da* " dv vv" A-" V
This equation is characterized by being homogeneous in x, y, y1
when these elements are given the weights i, «, n — i respectively.
Thus the differential equation
xy-y1'2 — yy + x = o
comes under this head; for giving x, y, y' the weights I, n, n — I respectively,
the separate terms have the weights i + 2 n + 2 n — 2 or 4 ;/ — i, 3 n + n — i
or 4 n — i, i respectively. These are equal to i if n = %. Hence the differen
tial equation is invariant under the group
d* by
VII. Uf= 4>(x) ¥• £ = o, rj = 4>(x). .'. T/ = <f>'(x), and // = .v,
//' = <j>(x)y' — <j>'(x)y. Hence the general type of differential equation
invariant under
Uf= +(X) is f[x, <}>(x)y' - #(X)y\ = o, or y' ~
This equation is characterized by being linear in y and y'.
Note. — Using the usual notation for the linear equation
y' + P(x)y=Q(x\
the group which leaves it unaltered is Uf=e~lrdt-^- The integrat
ing factor of § 12 is e!r'u. (Compare I', I. /)//. Eq. § 13.)
VII'. Uf=\l>(y)^-- It is readily seen that the general type of
ox
§ i9 DIFFERENTIAL EQUATIONS OF THE FIRST ORDER 57
differential equation invariant under this group is
i i//(y) \ dx i//(v)
v, , — *-)*•* x\ = Qt or - x = F(y\
y *w y •* »oo
This equation is linear in x and — •
dy
VIII. f.y=\f/(}') — • The general type of differential equation
invariant under this group (which includes III as a special case) is
In this equation the variables are separated.
VIII'. Uf=<l>(x) - • The general type of differential equation
ox
invariant under this group (which includes III' as a special case) is
y^(x) = F(y\
The variables are separable.
IX. Uf= 4>
dy
.-. V = 4>'(x)t(J) 4- ^ W^'O')/- Equations (37) are
dx_ dy dy'
o f(*]
/. u = x. u' may be obtained by solving the linear equation
in which x is treated as a constant. An integrating factor is -
58 THEORY OF-' DIFFHRKMTIAL EQUATIONS §19
Hence the ge neral type of differential equation invariant under
The transformation v = ( -^— reduces this to the linear equation
J
dx
Note
. — In particular, if ty(y) is y, Uf= <^>(jc)y * leaves unaltered
the equation /+; — -ry=fF(x). Hence the Bernoulli
equation ^-+Py= Qy is invariant under the group Cy=fel(s-})1'''£^-'
dx dy
J(l-^)/></x
The integrating factor of § 12 for this equation is - • (Com
pare EL Dif. Eg. § 14.)
IX'. Uf= *t>Mt(y) — • The general t\pc of differential cqua-
dx
.. . , ., . i dx \lf'(y] C dx 7V N
twn invariant under this group is - — = /"( v).
Considering y as the independent variable in this equation, the
latter is reduced to the linear form by the transformation /= I ' A •
J >
X. U=
u' is easily found by method (li) of § 18. The canonical variables
are dx dx .rv'-//i'
''
Hence the general type of differentia! equation invariant under
§i9 DIH'KRENTIAL KnUATlo.NS OF THE FIRST ORDER 59
Note, — Several particular cases are of special interest : —
i° If <b(x} — x\ the general type of differential equation invari
ant under
y
I lence xy1 — ny = _r*/M -^ - j is invariant under
uf~ x*-kx - -f
dx
The Riccati equation
dv
x —
conies under this head when n = 2 a ; for in this case
(Compare Boole, Differential Equations, p. 92 ; Forsyth, Differential
Equations, § 109.)
2° If <j>(x) = xrt n= i, the invariant differential equation reduces
to .vr' — v = .r1~r/M ~ ]• The right-hand member is simply a homo
geneous function of A- and y of degree i — r. Hence a differential
equation of the form v -— .vr' = .vA'/''f - ], where the right-hand incni-
\*/
her is a Jiomogeneous function of x and y of degree k, is invariant
dx dy
The integrating factor of § 12 is - — -• (Compare El. Dif. Eq.
60 THEORY OF DIFFERENTIAL EQUATIONS §19
3° If 4>(x) = xr, n = — i, the invariant differential equation reduces
to xy' + jr = x~l~rF(xy), or xy' + y =yl+rl?(xy). Hence a differential
equation of the form xy1 -\- y = ykF(x}) is invariant under the group
The integrating factor of § 1 2 is — - — - , a well-known fact.
X'. £7 = i/r 0') f x ~- + ;y; y; J • 77/<f ^«<?r«/ /v/te <?/ differentia
equation invariant under this eroup is xv' — nv = —r-rF[ — ]•
•AC;1) \**y
i° If \l/(y) =}'*, this differential equation reduces to
2° If i//0') =}'*, n= i, the differential equation takes the form
rN
Hence a differential equation of the form
y — xy' =y' [a homogeneous function of x and y of degree k~\
is invariant under the group Uf=\*~k( x - — Vyjry
3° If ;;=—i, ^(y}=y*, the differential equation reduces t
xy' +y = y'xg+lF(xy). Hence a differential equation of the form
dv
§i9 DIFFERENTIAL EQUATIONS OK THE FIRST ORDER 6l
The student should show that the following groups leave the corre
sponding differential equations unaltered :
XI.
\
XII. Uf= a ~ + b-, y = F(bx - ay).
ox oy
using method (r), § 1 8,
,,„'
'-A
x — vv'
using method (£), § 1 8.
fdi
XIV. 67=<£(
XIV. Uf=ti(
xv. iy^ ±r r.
dx 7 dy
xvi. uf= ^
«,WO*y 4- A+ v') =
where M = ^'^ v= C^(x)itx.
* This group is characterised l>y liaving ^ a function of \ only, and 77 a linear func
tion of y. It is mentioned by 1'rotes.sor Dickson, />'////<•//// of" the An:. Math, .Si'< .,
Vol. V. p. 453.
62 TIIKORY OF DIFFFKFXTIAL EQUATIONS §19
XVI'. Uf= <M
where
S***1 'Xv, o- EEj
Remark. — When a differential equation is recognized as coming
under several of the above heads, and the corresponding integrating
fastors are distinct, the solution of the differential equation is obtained
at once by equating the quotient of two distinct integrating factors
to an arbitrary constant (§ 16).
Thus the differential equation
is linear. Hence, from VII, the group Uf=x-J~ leaves it unaltered,
i dy
and gives the obvious integrating factor — •
But it is also readily seen that each term of the equation is of the
weight ;- when x, y, y' have the weights i, r, r — i respectively ; hence,
from VI, the group Uf=x—-+ry {- leaves the equation unaltered,
dx dy
and gives the second integrating factor - — • The solti-
(r— i>v>' — x
tion of the equation is therefore"
/ Ny ,. ,
=(r — i V- — x = const.
x
It may be noted that the equation also comes under X, 2°, and is
erefore invariant under Uf==^
iL
previously found integrating factor -
therefore invariant under Uf=x- ' • +xl~ry — • This leads to the
(Av dy
§§i9)2o DIFFERENTIAL EQUATIONS OF THE E1RST ORDER 63
' *
As another illustration of a class of equations obviously invariant
under several distinct groups, the equation
•*// — /+1 = xr or xy' - y = -
may be mentioned. Under the head of VI it is readily seen to be
invariant under Uf= (r-f \}x-+--\- ry-— : as a Bernoulli equation, IX,
dx dy
Tr+l fif
it is invariant under Uf=* — • From these its solution is found
y ty
at once to be
» ? =
This equation also comes under X, 2°.
20. Second General Method for Solving a Differential Equation.
Separation of Variables.* — The simple form of the differential equa
tions invariant under the group of translations 6^=— (I, § 19) sug
gests as a practical method for solving a differential equation invari
ant under a known group the introduction of canonical variables
(§ 10). The reduction of the group to the canonical form reduces
the differential equation to the form
in which the variables are separated. The solution is then obtained
by the quadrature ,-•
y = J F(X)JX + c.
Finally it is necessary to pass back from the canonical variables to
the original ones.
* This method was discovered by Lie in 1869, thus antedating the method of J 12 by
live years. 1 listorically it is ot interest because, it is the first known method of integra
tion which makes use of the invariance ot" a differential equation under a group.
64 THEORY OF DIFFERENTIAL EQUATIONS § 20
Since the differential equation invariant under Uf=. — *( I', § 19)
WP
is of the form — ^— = dx, the reduction of the group, under which a
differential equation is invariant, to this form also enables one to
separate the variables in the differential equation.
While either of the above transformations brings the differential
equation into a very simple form, the actual introduction of canonical
variables into the differential equation and the final passing back to
the original variables may not prove as simple as in the case of other
variables that could be used to equal advantage. Thus, for example,
if, in the group £7= £^ + 17-^- which leaves the differential equation
uoc ^y
unaltered, £ is a function of x only, the introduction of the new vari
ables (§ 9)
x = x, y = u(x,y)
reduces the group to the form
whence the differential equation must take the form (VIII', § 19)
(40) t(x)y' = F(y\
in which the variables are separable at once.
This set of variables works especially well in the case of two perfectly well-
known classes of differential equations, and leads to the usual methods for solv
ing them :
i° The homogeneous equation
Mdx + Ndy = o,
* Owing to the complete symmetry of the two groups fTf--- • and Uf~- , we-
dx fly
shall sny that the group in either case is in the canonical form, and the variables that
reduce a group to either form will be said to be the canonical variables of the group.
§20 DIFFERENTIAL EQUATIONS OF THE FIRST ORDER 65
where M and TV' are homogeneous and of the same degree, is left unaltered by
the group (IV, Note, § 19)
Uf=lXg.+yV.
d* dy
V
The new variables x = x, y —- — reduce the group to the form
whence the differential equation assumes the form (40), an-d the variables are
separable. (Compare EL Dif. Eq. § 10.)
2° The equation
- o
is left unaltered by the group (V, Note, § 19)
Hence, the new variables x — x, y — xy reduce the equation to the form (40)
in which the variables are separable. (Compare EL Dif. Eq. § 12.)
In an analogous manner, if rj is a function of y only, the introduc
tion of the new variables
x = u(x,y), y=y
reduces the group to the form
whence the differential equation must take the form (VIII, § 19)
in which the variables are separated.
More generally, if <f>(x) and \l/(y), any functions of the respective
canonical variables, are taken as new variables, it is readily seen that
the resulting differential equation will have its variables separated.
In certain cases such forms can be chosen for these functions as to
simplify the actual work required in introducing new variables.
66 THEORY OF DIFFERENTIAL EQUATIONS §§ 20, 21
Remark. — It is interesting to note that the knowledge of a group,
under which a given differential equation of the first order is invari
ant, enables one to find both an integrating factor (§ 12) and a set
of variables which are separable in the transformed equation. (Com
pare El. Dif. Eq. § 17.)
The integrating factor can be written down at once when the dif
ferential equation has been solved for ^-, or what is the same thing,
dx
when it has the form M dx -\- N dy = o.
To find the new variables that are to be separable, the solution of
another (frequently simple) differential equation of the first order
(giving the path-curves of the group) and usually one or several quad
ratures are necessary.
In actual practice, neither method should be insisted upon to the
exclusion of the other. In Table I of the Appendix will be found a
list of the more commonly occurring and easily recognizable classes
of equations of the first order, and methods for solving them.
21. Singular Solution.* — Let
(25) /(*,.?, y>=o
be an invariant differential equation under the non-trivial group
Its* family of integral curves being left unaltered, as a whole, if this
family has an envelope, the latter must be an invariant curve of the
group ; moreover, it is a path-curve, since the group is supposed to be
non-trivial, thus interchanging the integral curves among themselves.
The equation of the envelope being a singular solution of the dif
ferential equation (EL Dif. Eq. § 30) the value of its slope/ -at each
* Tli is section is based on an article by |. M. Page, entitled " Note on Singular
Solutions" in tin; American Journal of Mathematics, Vol. XVI II, p. 95.
§21 DIM KKKXTIAL K.nrATlONS OK THE KIKST ORDER 67
point (xfy) must satisfy (25). Since the slope of a path-curve at the
point (x, y) is ^ • *- /, the equation of the envelope must be con
tained in * "f? ;>'
(41)
Remark. — In the above process (41) was found as the equation
of a path-curve which satisfies the differential equation. If a par
ticular integral curve happens to be a path-curve of the group, its
equation is also included in (41). But all extraneous loci, ^uch as
nodal, cuspidal, and tac-loci (/?/. Dif. Eq.y § 33) which may be path-
curves but are not solutions of (25) will not be included in (41).
Ex. 1. xy-y's — jrV + x = o.
This equation is invariant under Uf=2x-~ — \~)'"^ , 0% § £9)-
Its general solution is s-x2 — cf + i = o.
Replacing v', wherever it occurs in the differential equation, by
- gives lT(4 x* — y4) = o.
x = o is a particular solution for c = oo .
4 x2 — j'4 = o is the singular solution.
Ex. 2. (i+A:V2=i-
This equation is invariant under Uf=:~. (I, § 19.)
y = - = oo. In this case, writing the differential equation in the form
— = o gives the singular solution i -f-lr2 = o.
Kx. 3. .xy-'-.vr'-v-o. (VI, § 19. « = -2.)
'I-:x. 4. r/;rr''-' - 2 .vr' +.v = o. (IV, § 19.)
68 THEORY OF DIFFERENCIAL EQUATIONS §21
Ex. 5. /3 — 4 ATI-' + Sy2 = o. (VI, § 19. n = 3.)
Ex. 6. v = 2 AT' +/y3. (VI, § 19. w = «.)
Ex. 7. xy- + ^'' +1 = 0. (VI, § 19. n = ~ J.)
It is suggested as an interesting exercise that the student examine,
in the light of the Lie theory as presented in this chapter, the vari
ous examples involving differential equations of the first order to be
found, for example, in Chapters II, IV, V of the author's Elementary
Treatise on Differential Equations.
CHAPTER III
MISCELLANEOUS THEOREMS AND GEOMETRICAL APPLI
CATIONS
22. New Form for Integrating Factor. — In § 12 it was seen that
i
V -n
R
is an integrating factor for
Mdx + Ndy = o
if the latter is invariant
under
FIG. i
Lie, by purely geometrical considerations, gave a new form * to this
factor, which is not only interesting but also useful in certain classes
of problems. In Fig. i, let
be some one of the integral curves of the differential equation. The
infinitesimal transformation of the group transforms this into an
infinitely near curve of the family
by transforming any point (x, y) of it into (.v -f- £ &a, y -\-
/'/', — the distance between these points is V£" + 17" &*.
First published in the Gesellschaft der Wissenschaften zu Christiania, 1874.
69
70 THEORY OF DIFFERENTIAL EQUATIONS §22
M
The slope of the tangent at P is -- . If T is the point (x — Nt
y + M), the length of/'7'is -\/M'-+~N*t and the area of the paral
lelogram PTRPi is (£M+rjlV)8a, or — .
P
Let 8;z = PN, the length of the normal to the first curve at P,
intercepted by the second curve ; this is, to within infinitesimals of
higher order than the first, equal to PQ, the altitude of the above
parallelogram. Hence
or
(2} =•-
=
This form of the integrating factor is serviceable in the case of an
interesting class of differential equations :
If the integral curves of a differential equation are known to be
^
a family of parallel curves* for which — is constant all along each
6a
one of the curves, it follows at once from (42) that
U»0
is an integrating factor. The involutes of a curve, which are the
orthogonal trajectories of the tangents to the curve, are known to
form a family of parallel curves. Hence an integrating factor of the
form (42') is known at once for their differential equation.
Ex. Find the involutes of the circltj x- -f- y~ = I.
The differential equation of the tangents to the curve is f writing/ for
* Two curves are said to be parallel, if the distance between them measured along
the normal to one of them is constant all along the curve. (In this case, it is well known
that the normal to either curve is normal to the other.)
§22 MISCELLANEOUS THEOREMS
Hence the differential equation of the family of involutes is
71
(xy -f -V
The integrating factor given by (42) is
— l d - o.
To integrate the exact equation
(.vr -f V** -f ^ -
one may proceed in the usual way (see El. Dif. Eq. § 8) to integrate
where ^' is considered a constant. Multiplying numerator and denominator by
x — y\fx* -f- yfi — I, this becomes
r .'••"• -
J ,\-- r _r'J
Letting .r2 -f r- = /,
= i gin_1 _
=r x/.r- + r- — I + sin'1 - — .
Hence, the equation of the family of involutes is
— I -f sin~! - — tan"1 * = const.
^ .,-' + ^2 JT
72 THEORY OF DIFFERENTIAL EQUATIONS §§ 23, 24
Remark. — From the nature of the problem, it is evident that the family of
involutes is invariant under the group of rotations &/= — y -. - + x -*£-• Hence,
the methods of §§ 12 and 20 are also applicable. It is readily seen that the inte
grating factor given by the method of § 12 is the same as that found in the text.
The method of § 20 should be carried out as an exercise.
23. Two Differential Equations with Common Integrating Factor.
If //, is an integrating factor for two distinct differential equations,
= o> and M2 dx + N2 dy — o,
_ = o and
dy dx dy dx
^ 1°S> _ M d]o£f' = **M±
dx dv dy
(A<>\ .•
a lo
.
I O.T dy dy djf
Here N^M^ — N.M^^Q, since the differential equations are sup
posed to be distinct. Hence (43) can be solved for - — sL£ and
dx
— 2!L£. • log/x can then be determined by a quadrature, and /x may
dy
be obtained at once from this. Hence the
THEOREM. — If hvo differential equations of the first order are
known to have a common integrating factor, the latter can be found
*
by means of a quadrature.
24. Isothermal Curves. — A family of curves which, together with
the family of orthogonal trajectories, divides the plane into infini
tesimal squares, is called a family of isothermal curves. In general,
* This is also obvious from tin- form of the differential equation, when cleared of
fractions, viz. : x + yp — Vi -f /-. (Sec II, Note, $ 19.)
MISCELLANEOUS THEOREMS
73
FIG. 2
a family of curves and their
orthogonal trajectories divide
the plane into infinitesimal
rectangles. For, selecting
any pair of neighboring
curves, /and //(Fig. 2), of
the one family it is always
possible to find a pair, A
and B, of the second family
to form an infinitesimal
square * with them ; besides,
selecting any third curve ///
of the first family, a fourth
curve IV can be found such
that A, /?, ///, IV form a square also ; again, selecting any third
curve C of the second family, a fourth curve D can be found such
that C, Z>, /, // form a
square. But with these se
lections made, the curves
C, D, III, IV do not, in
general, form a square.
Concentric circles are read
ily seen to be isothermal curves.
Their orthogonal trajectories
are the straight lines through
the common center (Fig. 3).
Any pair of circles of radii
r and r + \r respectively
(r>o) form an infinitesimal
FIG. 3 square with any two of the
* This curvilinear quadrilateral is a square when infinitesimals of higher order than
the first arc neglected, the length of arc of one of the sides being taken as an infinitesi
mal of the first order.
74
THEORY OF DIFFERENTIAL EQUATIONS
§24
straight lines which intercept the length Ar on the inner circle. Moreover these
same two lines form squares^with any other pair of circles of radii kr and
^(r-fAr), respectively, k being any constant different from zero.
From the definition of isothermal curves, 8«(of § 22) can be made
the same, at any point, for this family of curves and for that of their
orthogonal trajectories. Moreover, if the differential equation of the
one family is
(17)
that of the other is
(17')
Hence the two equations have a common integrating factor, as is
evident from the form (42). To determine this integrating factor,
the method of § 23 applies. The equations (43) take the form
_ -- — — - — —
dy dy ax
log /x Nd log ^
dx
dy
dN
~dj"
dM
dx
whence
(44)
dJogjA
dx
dM
-^.
dy
-=
o
., ,T
M— -- N-~
dx
^
ox
M* + N*
dx
dx
Or
dy
dy
dx
Equations (44) are interesting, not only because they enable one
to find /A by a quadrature, but also because they lead to the condition
§24 MISCELLANEOUS THEOREMS 75
that the integral curves of the differential equation (17) be isothermal.
For, differentiating the first of (44) with r«pect to_y and the second
with respect to x, and equating
(45)
The general solution of this is *
(46) = tan [$(# -f iy) -f ty(x — iy)~],
where <£ and ^ are arbitrary functions.
The condition (45) is not only necessary that (17) be the differen
tial equation of a family of isothermal curves, but it is also sufficient.
For, when M and ^ satisfy (45), equations (44) are consistent, hence
a common integrating factor for (17) and (17') can be found. But
the sum of the squares of the coefficients of dx and dy is the same
for these two differential equations. Hence, remembering the form
(42), S// must be the same (to within a constant factor, which may
be made unity by a proper choice of neighboring curves) in the two
cases at any point. Hence the integral curves of (17) are iso
thermal curves.
Remark. — The condition for isothermal curves in terms of their
finite equation and that of their orthogonal trajectories is obtained
in Note III of the Appendix.
i° In the case of the family of concentric circles, x1 + y*- — const., the differen
tial equation is x dx -\- y dy = o. Hence (45) is satisfied, since y2 tan"1-^ = o.
While the solution of this differential equation, ,as well as that of the differen
tial equation of the orthogonal trajectories, y dx — x Jy = o, is very simple, it is
interesting to note that (44) give very readily
/I 2 x dx -f- 2 v dy ,, / o . ox
</log/* = -- y / ^ = -</log(>2+y*). .-. n =
This is the common integrating factor for the two equations.
* See El. Dif. Eq. \ go.
76 THEORY OF DIFFERENTIAL EQUATIONS §§24,25
2° The family of circles tangent to the axis of y at the origin x2 + y- — ex = o
has for differential equation ^c'2 — y'2)dx -f zxydy = o. It is readily seen that
2 JCV
V2 tan"1 - *• — = o; hence these circles form an isothermal system. The dif-
x--f
ferential equation of the orthogonal trajectories is 2 xy dx— (x2 — y~}<ty — o.
While this is easy to integrate, it is worth noting that (44) give /* — —
(xi +y2)2
Moreover, since the differential equation is " homogeneous," it is invariant under
the group Uf=x^-+y^L (IV, § 19). Hence, a second integrating factor is
dx dy
(§ 12) /io = — - . The solution of the equation is therefore (§ 16)
= const, or .r2 -f y- — cy = o,
/"2 y
the equation of the family of circles tangent to the axis of x at the origin.
— 2
Show that the following curves are isothermal, and find their
orthogonal trajectories :
Ex. 1. The equilateral hyperbolas xy — const.
Ex. 2. The similar conies ax1 4- by* = const., when and only when
b = ± a.
Ex. 3. The coaxial circles through the points (i, o) and (— i, o),
y
25. Further Application of the Theorem of §23. — An obvious
corollary of the theorem of § 23 enables one to find an integrating
factor, by means of a quadrature, for an interesting set of differential
equations. This corollary is : If the ratio of the integrating factors
of two differential equations is a known function, the integrating
factors can be found by a single quadrature. For, suppose that
(47) &=4>(X,y)
to
is a known function, where ^ and /x2 are the integrating factors of
J/! dx -f TVi dy •=• o and M.2 dx 4- N* dy = o
§25 MISCELLANEOUS THEOREMS 77
respectively. If the second equation be written in the form
•
<£ M., dx + $N» dy — o,
its integrating factor is also fi\- Hence it and the first equation have
a common integrating factor and, by the theorem of § 23, this can
be found by a quadrature.
Suppose, now, that it is known that the solutions of three differen
tial equations of the first order
J/! dx -\-JlVl{fy = o, M* dx + N» dy = o, M3 dx + Nz dy = o
can be made to assume such forms, fa = const., fa = const.,
fa = const., that
(48) fa = fa + fa.
If Hi, ft*, /*3 are their respective integrating factors,
dfa = n}(Mi dx + NI dy), dfa = ^(M^ dx + N.2 <fy),
Because of the identity (48)
dfa = dfa 4- dfa,
or
(49)
whence ^ = M^i + ^ = MI^ + ^ and
, By the corollary above, H\ can be found by a quadrature ; and /u;,
is then known from ('47'). After finding <£, and fa by a single quad
rature each, fa is given immediately by (48). Hence the
78 THEORY OF DIFFERENTIAL EQUATIONS §25
THEOREM. — If it is known that the solutions of three differential
equations of tJie first order can be put in sitcJi forms <£[ — const.,
<b., = const. , d>.j = const. tJiat ,
=
these solutions can be found by means of three quadratures.
This theorem has some interesting applications in the theory of
surfaces * :
A. If the rectangular coordinates of any point (xty, s,) on a sur
face are expressed in terms of the parameters u and ?', the expression
for the element of length of arc is, using the usual Gauss notation,
ds> = E dir + 2 Fdu dv + G dv\
where
W£Y+(£Y+(£Y, F=^* • 8v8)' • 8z dz
\vuj du dv
dit du \du ' d/ dv du d
The differential equation of the lines of zero length, usually called
minimal lines, is then
(50) E du* + 2 Fdu dv -f G dir — o.
This differential equation, being of the second degree, is equivalent
to the two _
E du + F+ ^F'2-EG dv = o
(51)
E du + (F- V/^2 - EG) dv = o,
which are essentially distinct, since it is always presupposed that
EG — //2 is different from zero. Let «(//, v) = const, and (3(u, v)
= const, be the solutions of (51). These are the equations of the
minimal lines. Choosing them for parametric curves, equation (50)
* These applications will be of interest to those only who have, at least, a slight
nrqiiiiintance with the elements of Differential Geometry. They have been taken from
Lie's I'orlesungen uber Differentialgleichungen, Chap. 9.
§25 MISCELLANEOUS THEOREMS 79
takes the form da tip = o, i.e. E(a, /3) = G(a, ft) — o, and the expres
sion for the element of length of arc is
</s-=2F(u, P) tin tip.
Introducing the new parameters i^ and ?\ defined by
where </> and «// are any desired functions of their respective
arguments, tf = *(„„,.,)(,/„? + W),
since tiatiB = (ih<\ + <h'\~)- This form of the expres-
V(*W(py *
sion for the element of length is characteristic of isothermal para
metric curves. (Compare Note III of the Appendix). Hence,
2 //! = U= </>(«) 4- \I>(P) = const.
and 2 ivi = V= <£(«) — \j/(fi) = const.
are the equations of the isothermal curves and their orthogonal tra
jectories, respectively. Since <£(«) = const, and ty(J3) = const, are
equally well the equations of the minimal lines, it is evident that the
identity (48) is satisfied by the equation of any isothermal system
and those of the minimal lines. It follows then from the theorem
above that the differential equation of a family of isothermal cinves
on any known f surface can be integrated l>\ means of quadratures.
Besides, the knowledge of a family of isothermal lines on a known
* In the case of a real surface, a and /3 may be selected as conjugate complex
functions of u ;,nd v, when the original parametric curves are real. Real isothermal
curves are then obtained by choosing 0 and \f/ conjugate functions of it. and /j
respectively.
t A surface is said to be known if the values of .r,^, z in terms of the parameters
«, -v are known, or if the forms of E, F, (7, and of />, D' , D" (to be introduced below)
are given in terms of//, v. In this particular case /:, F, G only need be known, mini
mal anil isothermal lines not depending upon D, /•>', D" .
8o
THEORY OF DIFFERENTIAL EQUATIONS
§25
surface enables one to integrate the differential equations of the mini
mal lines (51) by means of tu>o quadratures.
Remark i. — For surfaces of the second order, surfaces of revolu
tion, and minimal surfaces, the lines of curvature (see B below) are
known to be isothermal lines. Hence, in the case of these surfaces
the differential equation of the lines of curvature can be integrated
by means of quadratures.
Remark 2. — In the case of a minimal surface the asymptotic
lines are also isothermals. Hence, on such a surface the differen
tial equation of these lines can also be integrated by means of
quadratures.
B. The tangent plane to a surface at a given point cuts the sur
face in a curve which has a double point at that point. In general,
the directions of the tangents to the two branches of the curve at
that point are distinct. In this way two directions (in general) are
determined at every point on the surface. A curve on the surface
whose direction at every point coincides with one of these directions
is called an asymptotic line. So that, in general, through each point
on the surface there pass two asymptotic lines. The differential
equation of the asymptotic lines is
(5*)
when
j
>
d2x dx dx
du2 du dv
d'2y dy dy
dir du dv
d2z dz dz
du- du dv
D dir +
, />' =
2 D'du dv + D"(
d*x dx dx
fir = o,
d2x dx dx
du dv du d?>
d*y dy dv
du dv du dv
d2z dz dz
dv2 du dv
d2y dy dy
dv- du dv
d*z dz dz
du <>v du dv
dv~ du dv
In case £>D" — D'2 = o, the two curves coincide. This happens
at every point of a surface where the Gauss measure of curvature
is zero.
§25 MISCELLANEOUS THEOREMS 8 1
Another system of curves playing an important role in the theory
of surfaces is that of lines of curvature, which have the property, that
along them consecutive normals to the surface intersect. Their dif
ferential equation is given most conveniently in the determinant
form
(53)
drf — du dv dir
E F G
D D' £>"
= o.
This differential equation is again of the second degree, so that
through each point pass two lines of curvature. These are mutually
orthogonal, and besides their directions are harmonic conjugates
with respect to those of the asymptotic lines through the same point,
as may be seen readily from the forms 'of equations (50), (52), and
(53)-
Suppose that on a certain surface the asymptotic lines are known
to cut out rhombuses.* This can be expressed analytically in the
following way :
The selection of the asymptotic lines as parametric curves does
not affect the appearance of the expression for the element of length
of arc. But since u = const, and v = const, must then be the solutions
of (52), it follows that Z> = Z>" = o. Hence the differential equa
tion of the lines of curvature (53) reduces to
(53') Edir- G<h>- = o.
The elements of length along the parametric curves are ^/ E du and
^J G dv. These will be equal at every point on the surface, and the
surface will therefore be divided into rhombuses, if V E = A(//, i')4>(N)
and ^/ G = X(//, 7')^(7'). (See corresponding argument in the
case of isothermal lines in Note III of the Appendix.) Letting
* This is known to be the case for surfaces of constant Gauss curvature, for
example.
82 THEORY OF DIFFERENTIAL EQUATIONS §25
j <£(//) du = U, I \l/(i>) dv = F, the expression for the element of
length takes the form
The differential equation of the lines of curvature takes the form
dU~-dV- = Q ;
whence the equations of the lines of curvature are
U + V = const, and U — V — const.
Since the identity (48) holds, it follows that if the asymptotic lines
divide a surface into rhombuses, the asymptotic lines and lines of
curvature can be obtained by means of quadratures.
CHAPTER IV
DIFFERENTIAL EQUATIONS OF THE SECOND AND HIGHER
ORDERS
26. Twice-extended, n-times- extended Group. — A transformation
of the variables .v andjy carries with it a transformation of the various
derivatives of y with respect to x. Thus, just as the point trans
formation
*i = $(*of)> ***1fftJ)
carries with it (§ 13)
dx By
so it also implies d*^
dx dy
The transformation
*! = <i>(x, y), }'i = ^(^,y), y\ = x(x> y> y)> }'i" — <*(x> y> y'> y")
affecting the four variables x, y, y',y" which is implied by the point
transformation is known as a twice-extended point transformation*
Starting with the one-parameter group of point transformations
( i ) #! = <f>(x, y, a), )\ = if/ (x, y, a},
* In precisely the same way we arc led to the n-times-extended tr ana formation
— = e(x,y,y'ty", •••,/")).
84 THEORY OF DIFFERENTIAL EQUATIONS §26
by employing the method of reasoning in § 13, the corresponding
twice-extended transformations
(54) x\ = <K*> y> a)> }\ = $(x> y> a\ yi = V ' = x(-v» }'> /> <0>
are seen to constitute a one-parameter group in the four variables
x, y, y', y". This group is known as the twice-extended group corre
sponding to (i).
Writing as the symbol of the infinitesimal transformation of the
twice-extended group
(55)
where as before
77", which is • --, may be found in exactly the same way as ?/ was;
§a
thus
77 SffSp ^v ^v- ^r dx dx
Reasoning as before we have the n-times-exUnded group
= 4>(x, y, a), yl = $(x, y, a,), yl = 'j± = \(x, y, }•', a),
"x\
§ 26 DIFFERENTIAL EQUATIONS OF THB SECOND ORDER 85
the symbol of whose infinitesimal transformation may be written
where
Remark. — While >/' is a quadratic polynomial inj' ([24], § 13), it
is seen, on expanding (56),
(58) ,"=^v
djc 5)'
that ?;" is linear in y". In the same way ry(*) is seen to be linear in
y(k) for k > i, since
__
By dy'
In I, Uf, f = o, r;-i. /. 17' = o, T;"E:
«r
Hence, U^f=^--
dy
In II, Uf=-y-f + xd/t $==-}>, -n = x
O-^ Cy'
v"=3/'2 + 4yy", iiiv=s(2yy"
Hence,
in in, u/mf-t ^ = 0, 77=^. .-. r = v', 17" =/', •••» i?(w)=y"v.
^^+y:+," + ...+y"».
86 THEORY OF DIFFERENTIAL EQUATIONS §§ 26, 27
In IV, Uf=x&+y&t £ = x, i,=y. .-. ,' = o, i»"=-/', r/"EE-2/",
Extend the following groups :
Ex. 1. f • Ex. 2. xf. Ex. 3. ^-j
d,r . ox dx " d
Ex. 4. a* + ^ Ex. 5. 0(. Ex. 6.
5jf J dy Jdx
Ex. 7. ^.
dy
27. Differential Equation of Second Order Invariant under a Given
Group. — The effect of any transformation (i) on the variables x and
y is to transform the differential equation
(60) /(*,;',/,/')=<>,
by the corresponding extended transformation (54). In order that
the equation (60) be invariant under the group (54), it is necessary
and sufficient that ([12], § n)
(61) £/"/= o whenever /(*, y, y', y")=o.
Using the same argument as was employed in § 18, it is seen that
all the differential equations of the second order invariant under the
group are obtained by equating to zero an arbitrary function of three
independent solutions of ([9], § n, footnote)
§27 DIFFERENTIAL EQUATIONS OF THE SECOND ORDER 8/
Passing to the corresponding system of ordinary differential equa
tions
dx dy dy1 _ dy1'
the first three members are seen to be the same as those of (37),
§ 18. Hence, two of the solutions, u(x, y) = const, and u'(x, }',}>')
= const., may be found by the methods of that section.
To find a third solution, u"(x, y, /, y")= const., which must neces
sarily involve y", use may be made of the two already found to
eliminate j> and y' from -^- =^-, or x and/ from -]' = — , or x
dx £ dy TJ
and y from -^- = 2_ (whichever turns out to be the simplest). Each
dy 77
of these differential equations is linear since rj" is of the first degree
in/' (§ 26, Remark). This linear equation can be solved by means
of two quadratures. (See El. Dif. Eq. § 1 3).
Lie has given a most ingenious method for finding a form for
//'i.v, yf y,1 y"), without any integration whatever when u and u' are
known :
Consider the differential equation
(64) //'(*, _>',/)-« «(*,.)') = &
where a and ft are constants. Since u and //' are invariants of the
once-extended group U'f, (64) is invariant under the group Uf\ that
is, its integral curves are interchanged among themselves by the
transformations of this group. Keeping a fixed, an invariant family
of a single infinity of integral curves corresponds to each value of ft.
Still keeping a fixed and allowing ft to take successively all possible
values, an infinity of such families, constituting a double infinity of
integral curves, is determined by (64). This larger aggregate is in
variant under the group Uf, since each of the constituent families
corresponding to the same value of ft is. It is evidently the set of
88 THEORY OF DIFFERENTIAL EQUATIONS §27
integral curves of the differential equation of the second order ob
tained by differentiating (64), thereby eliminating ft ; viz.
du du< Bx
,,.
(65) - -- « -— — o, or -— =
dx dx du
— a
Since its integral curves are interchanged among themselves by
every transformation (i), it is invariant under the group Uf. Hence,
by(6l) fM \ du'
Un( - — a } = o whenever — = a.
\;/// ) du
But a being a constant, C7"l — — a ]= U"( — •] ; i.e. it is indepen-
V du J \dn
/ J l\ \ /
dent of «. £/"[ - — ) is therefore identically zero ; which is sufficient
, \d*/
to make — an invariant of (54), ([9], § n).
du
Since //' contains y' (§ 18), — ^ o, and - must contain y"*
ay' du
Hence, -— = const, may be used as the third solution of (63). The
du
general solution of (62) may then be written in the form
(66) f(u, «', ^) = o, or ^' = F(u, »').
\ du J an
This is the general form of the differential equation of the second
order invariant under the group Uf. We have therefore the follow
ing most important
THEOREM. — If f(x, r, y', _r") = o is a differential equation of the
second order invariant under the group i'f^ and if u (x, y) is any
* An invariant of the extended group U"f which involves j" is known as a second
differential invariant of the group Uf.
t Attention should be called to the fact that while every differential equation of the
first order is invariant uivk-r an indefinite number of groups (see \\ 15, 17) a differen-
§27 DIFFERENTIAL EQUATIONS OF THE SECOND ORDER 89
invariant and //'(x, v, v') is any first differential invariant of Uf, the
introduction of the ncu> rariablcs
(67) x= it (x, r), y = u'(x, y, y')
reduces the differential equation to the form
(66') '^ = F(X,y),
which is of the first order.
In actual practice the introduction of the new variables is usually
most readily effected by noting that
Qy 4- ^ v' 4- ^ v"
<ty_dx d)> dy'J
dx dx dx ,
dx dy "
is some function of u = x, u' = y, and //". When this function is
obvious upon inspection, //" can be determined in terms of x, y, ' •
In other cases it may be necessary to solve
dy dx dy dy' "
dx dv
fory,y',y" in terms of JT, y, '^, x. Substituting these in the differ-
dx
ential equation, A* must disappear, and the resulting equation must
take the form (66').
After having solved (66'), its solution
(68) 4>(u,t/',f) = o
a differential equation of the first order. Rut owing to the inva-
riance of u and u' (68) is invariant under Uft so that it may be
)lved by the method of § 12 or that of § 20.
il equation of the second (or higher) order is in general not invariant under any
group. (See Note IV of the Appendix.) On the other hand, a large number of them,
including most of the known forms, are, and these will be considered in this chapter.
90 THEORY OF DIFKKKKNTIAL EOl'ATIONS §28
28. Illustrations and Applications.
I. 6^" = -" (. £ = o, yj = i. .'. r)' = o, -r}" = o (§ 26). P^quations
(6i\ nrp 7 r 7 ri
\V6J ^lc ^v _ //v _ <ty _ ay
o i o o
w" = y. Hence, the general type of differential
equation of the second order invariant under Uf= •— is f(x, j1', JIP") = o
0ry" = F(xty'). This equation is characterized by the absence of r.
Note. — The transformation of variables x=xt y = }>' (§ 27) re
duces the differential equation to
(66') & = f(x,y).
ax
This is precisely the usual method for solving an equation of thus
type. (See El. Dif. Etj. $ 57). Solving the solution of (66') for y,
it takes the form
in which the variables are separated, as must be the case (I, § 19),
since this equation is invariant under the same group (§ 27).
I'. Uf= |P£; It is readily seen that the general type of differen
tial equation of the second order invariant under this group is
/(y, _y', y") — o, ffry" = F(y, y'). This equation is characterized by
the absence of .Y.
Note. — The transformation x=y, y=y' (§ 27) reduces the dif
ferential equation to one of the first order (66'). Its solution
ffy
y=/(x, <•), or ^ =f(y, c]
is a differential equation with A* absent again, as must be the case
(If, § 19 and § 27). This is also the usual method for solving an
(•([nation of this type. (See El. Dif. Eq., § 58.)
§28 DIFFERENTIAL KoUATK )\S OF TIIK SITCOM) ORDKK 91
Remark. — Owing to the simple form of an equation invariant
under either of the groups Uf = ~ - or £^"= ---, it is frequently
desirable to introduce canonical variables in case a given differential
equation of the second order is known to be invariant under some
group. When the introduction of canonical variables is not prac
ticable, other changes of variables reducing the group and equation
to known forms may prove desirable. (Compare § 20.)
(§ 26). Equations (63) are
(6.3') *-
v — .vr'
.-. u = .\-+r, «'=• T ;;/ (§ 19)- Using the last two members of
equations (63'), //" = — -- — — • Hence the general type of differential
^ ' t\f *i f
1 of tlie second order invariant under Uf= — V-= + .v ; is
v,,, , - fo fy
Note. — The form of this differential equation is obvious from
geometrical considerations, since // is the square of the radius vector
to any point on an integral curve, //' is the tangent of the angle be-
twivn the radius vector and the tangent to the curve, while u" is the
square of the curvature, all of which are left unaltered by the group of
rotations about the origin. (Compare § 29.) In order to integrate
such an equation the method of I', Remark, requiring the introduc
tion of canonical variables (polar coordinates in this case) will usually
be found desirable.
Making use of the fart that ' and • are also first
VI +.v'"' Vi -r\'''2
differential invariants of the group of rotations (II, Note, § 19) other
92 THEORY OF DIFFERENTIAL EQUATIONS §28
possible forms of the invariant differential equation of the second
order are
I ' \ ' '** / ' \
~, - j and -- — ^ = F\ *-+y2, —
ViH-v'v (I+J'")< V Vi + y'v
III. Uf~y^- £ = o,r)=y. .•.ri'=y')ri"=y"(^26). Equa-
_ _
o y / /'
dx dy
o y
y' v"
.'.// = jc, «' = '•-, u" = - — * Hence ///<? general type of differ en-
r\f
tial equation of the second order invariant under Uf = y~- is
This equation is characterized by being homogeneous in y,y',y".
It is evident, at once, that an equation of this type is left unaltered
by the affine group Uf = y-^, since the finite transformations of the
extended group are x^ — x, y\ = ay, y^ = ay1, y" = ay".
Note. — An interesting equation of this type is the homogeneous
(or abridged) linear differential equation
The transformation x = x, y = ~ (§ 27) reduces the equation to
ax
a Riccati equation. (Compare EL Dif. Eq. § 73, 6°).
* The Lie method of vv 27 tjivcs u" — '1' , -' •' f-J , and the dif-
</// r- v ^y '
tcrrntial c(|uation —(' \ + /' M', ), \vliich is, of course, the same in form as
that lound in the body of the text.
§28 DIFFERENTIAL F.ni 'A I K ).\S OK THE SE<O.\I) ORDER 93
VI. U/=x + ny-. t = x,i, = ny. /. V = (« - i)/,
dx By
rj" = (n — 2)}'". Equations (63) are
x ny (n— i)/ (n — 2)/'
Hence the general type of differential equation of the second order
invariant under Uf = x — -f- nv ~— is f ( — , — — , - — ) = o.
ox ' dy \xn xn~l xn~-J
This equation is characterized by being homogeneous in x, y, y', y"
when these elements are given the weights i, n, n — i, n — 2 respec
tively.
Note. — Boole called an equation of this type homogeneous, and
gave as a method for solving it the transformation x = log x, y = ^-.
(See Boole, Treatise on Differential Equations, p. 215 ; Forsyth,
Treatise on Differential Equations, § 55). The new variables in this
transformation are a set of canonical variables. (Compare I', Remark.)
III'. Uf=.\— is a special case of VI. Here ;/ = o, and the
ox
invariant differential equation is of the form f(y, xy', x1)'") = o.
This equation is homogeneous in x. y', y" when these elements
have the weights i, — i, — 2 respectively ; the weight of y being zero,
the manner in which this variable enters plays no role.
III. Uf=L\^- may also be looked upon as a special case of VI,
corresponding to the value ;/ = oo. Boole deduced a special method
for this case (see Boole, p. 220; Forsyth, § 55) which is exactly that
of § 27 for this case.
IV. Uf=x — -\- v--- is the special case of VI for n — i. The
d\ - dy (
invariant differential equation is of the form f(^ , r', .vv") = o.
\ oc
94 TIIKORV OF niFFKKKNTIAI. KOUATK >\S §28
V. Uf=x <£* — y — is the special case of VI for n — — i. The
dx dy
invariant differential equation is of the form f(xy, x'-y', .ry") = o.
vii. u/= *(*)§£• t = o, -n =
It is readily seen that
Hence ///<? general type of differential equation of the second order
-^-
invariant under Uf=$(x}-^- is f(x, <j>y' — $'}', <f>y" — <j>"y} = o, or
Note. — An interesting equation of this type is the complete linear
equation
(70) /' -f />(*)/ + Q^)y = X(x),
which is obtained from the general form by letting F be linear in
7/EE <£/ — <£'v. Bearing this fact in mind, it is clear that y = <£(.v)
satisfies the abridged equation (69), obtained from (70) by replacing
X(x) by o. Conversely it is readily seen (and will be left as an exercise
to prove) that \fy=y0 is a solution of (69), (70) is invariant under the
group Uf=\\) -. The transformation x = x, y=y{ty' — yjy (§ 27)
reduces the equation to the linear equation of the first order
(71) '
This property of the complete linear differential equation of the
second order of reducing to one of the first order by a transforma
tion that is known when a particular integral of the corresponding
abridged linear equation is known is not new. (See 7v7. Dif. Rq.
§ 53, i°.) The transformation employed above yields an equation
§28 DII'KKRKNTIAL EQUATIONS OF TIIK SKCOND ORDER 95
bearing a more striking resemblance to the original equation than
the transformation,
usually employed. The new variables in this transformation are a set
of canonical variables (I', Remark).
Other groups whose invariant differential equations are readily
found are the following :
VIII. tf/EEitfy)/-. f(x/-S-r -•>-?- =0.
VIII'. Uf= *(*) f(y, </>y, 4r>" + Wy') = o.
X. Uf=
XII. Uf=a + l>.
In Table II of the Appendix will be found a list of the more com
monly occurring and readily recognizable classes of equations of
higher order than the first invariant under known groups.
Ex. i . xyy11 4- -*y2 —yy' — o.
This equation is invariant under the group IV: £^"= x~-\- y-fr*
Introducing the new variables
*='[,> y=y'
96 THEORY OF DIFFERENTIAL EQUATIONS §28
the equation takes the simple form
^+? = o.
dx X
vv'
Integrating xy = a, or •*— = a.
oc
Integrating again ax1 —f = b.
Note. — Inspection shows that this equation is also invariant under
III: Uf=y^t and III': Uf = x-t>
dy ox
Ex. 2. (.r- +f-)y" + 2 (y - */)(i + /2) = o.
This equation is invariant under \he group II :
Introducing the canonical variables (in this case, polar coordinates)
the equation takes the form — = -f- y = o.
tbr
Here the independent variable is absent, but, instead of using the
•method indicated by the general method of § 27, it will be simpler
to solve this linear equation with constant coefficients by the usual
method for such an equation. (See EL Dif. Eq. § 45.)
y = a cos x -f b sin x,
To pass back to the original variables, multiply by y, whence
§§28,29 DIFFERENTIAL EQUATIONS OF THE SECOND ORDER 97
Note. — This differential equation is also invariant under IV.
Ex.3, x2}'}'" — (xy'—}')- = o. (Invariant under III, IV, . . . .)
Ex. 4. ji'y" + to'' - J')2 = o. Ex. 5. x'2y" = xy' - y.
Other equations invariant under known groups appear in §§39
and 40.
29. Further Applications. — Besides being able to recognize a
-roup under which a given differential equation is invariant from the
characteristic properties given in § 28
and enumerated in Table II of the
Appendix, it is possible at times, to
find such a group from the nature of
the problem giving rise
to the differential equa
tion. As examples, the
following may be noted :
i° The group of rota
tions about the origin
zs r\s
leaves unaltered
dx dy
R = the radius of curvature of a curve at any point,
p = the radius vector to any point on the curve,
r = the radius vector to the centre of curvature,
the distance from the origin to any line (such as the tangent
or normal) connected with the curve, thus OM and ON,
/W=the polar subtangent, = ON,
PN= the polar subnormal, = OM,
i/' = the angle between the radius vector and the tangent,
the remaining angles of the triangle OCP.
Hence a family of curves defined by a relation between any or all
of these is unaltered by this group ; the differential equation of the
98 TIIFORY OF D1FFFRFXTIAL FOUATIONS §29
family is therefore invariant under it. Passing to polar coordinates
(the canonical variables) will usually be found desirable in this case.
2° The similitudinous group Uf^x — -f- y — leaves unaltered
civ dy
0 = the angle between the initial line and the radius vector,
T = the angle between the initial line and the tangent to the curve,
<£='the angle between the initial line and the radius vector to the
centre of curvature,
<// = the angle between the radius vector and the tangent to the
curve,
the ratio of certain lines connected with the curve, such as
radius vector, radius of curvature, radius vector to the centre of curva
ture, intercepts of the tangent, normal, or of the curve itself, sub-
tangent, subnormal, length of tangent or normal from a point on the
curve to one of the axes, and the like.
Hence this group leaves unaltered the differential equation of a
family of curves defined by a relation between any of the above in
variant configurations. Passing to canonical variables, or to polar
coordinates (thereby reducing the group to III') may simplify the
problem of solving the differential equation.
3° Certain configurations could be enumerated as invariant under
the groups of translations Uf = ~ and Ufm4~* But as in either
case one of the variables is absent in the resulting differential equa
tion, the latter will suggest the group without considering the defini
tion of the integral curves.
Ex. Kind the family of curves for which the radius vector to any
point of a curve is perpendicular to the radius vector drawn to the
centre of curvature of the curve at that point.
The differential equation of this family must be invariant under
the group of rotations 11 and also the similitudinous group IV.
§§29,30 DIFFERENTIAL KOUATK >\S OK IIKlIlF.k OkDKR 99
Noting in Fig. 4 that the triangle POC is right-angled at O,
= cos, or = s
Hete P = Vxf+'f, R = , tan «// =— : - Hence the
/' * +yy
differential equation is
(.v2 +/)/'- (i +y*)(y-xf) = °-
30. Differential Equation of Order Higher than the Second Invari
ant under a Given Group. — The method of § 27 can be extended
without change to differential equations of higher order :
A differential equation of the ;/th order
(72) /(*,;',/,/', •••>j("))=°
is invariant under the group Uft if and only if
(73) U(n)f=o whenever/^ o.
All the differential equations of the «th order invariant under the
group are obtained by equating to zero an arbitrary function of ;/ -f- i
independent solutions of
These independent solutions may be obtained from the corre
sponding system of ordinary equations
dx f/v_,/r' </v"_ <&*>
\15) T — '— , — ' TT — "' = -'~rrr*
It was seen in § 27 that if //(.v, v) is an invariant of (Y, and
i/'(.\\ v, v') is a first differential invariant, then ' is a second differ-
(/!/
ential invariant. Hence,
(76) *£-<&* ft
100 THEORY OF DIFFERENTIAL EQUATIONS §30
is an invariant differential equation of the second order for all values
of the constants a and ft. Its integral curves constitute an invariant
family of oo2 curves. The <x>1 differential equations of the second
order obtained by keeping a fixed and giving to ft all possible values
have for integral curves oo1 such invariant families of oo- curves.
Grouping all these curves into one aggregate of <x>3 curves, this aggre
gate is invariant under the group since each of the families is. The
differential equation of this family is, therefore, invariant. It is ob
tained by differentiating (76), thus eliminating ft,
( \ —(^\— ^_ d~11' _
{77) dx\du) adx~ ~dtf
In order that (77) be invariant, we must have from (73)
TJ.,.(d"u{ \ , (/-//'
U — a •= o, whenever = a.
\ dir J dir
"Rnf- /"/'"
u i , ./ lv I — ^ ~T~9 '
dir J dir
i.e. it is independent of a. Hence, if (77) is to be invariant, £/"' — —
72 I ****
must vanish identically. So that - - is a solution of (74). Since it
atr
contains y (as may be seen readily), it is independent of //, //', — «
dlt
In the same way it can be shown, step by step, that a set of inde
pendent solutions of (74) is
u' du- <w </n-v.
du dir
Hence the general type of differential equation of order ft invariant
under the group Uf is
§30 DIFFERENTIAL EQUATIONS OF IIKUIER ORDER IOI
We have then as an extension of the theorem of § 27 the following
THEOREM. — If f(x> }', _v', y", •-, y(H)) = o ls invariant under the
gn>itf> L'f, ami ////(.v, r) /v an\ invariant, and //'(.v, v, r') is any first
differential invariant of Uf, t/ie introduction of the new variables
(79) * = M(-V> y\ y = "' (x> y> y')
reduces the differentia] equation to
u>)iich is of order n — i.
After having integrated (78'), its solution
is a differential equation, also invariant under Uf, since u and //' are.
Hence it may be solved by the method of § 1 2 or of § 20.
Many of the arguments of § 28 can be used here, almost without a
single change. Consequently, the results only will be given, it being
left as 'an exercise for the student to fill in the steps.
I. The general type of differential equation of the nth order
invariant under Uf=~~ is/(x, y', y", •••,/"))=o, which is charac
terized by the absence of y.
The transformation y=y't reducing the differential equation to one
of order n — i constitutes the usual method for solving an equation
of this type. (El. Dif. Eq. § 57.)
I'. The general type of differential equation of the nth order
invariant under Uf=^ is /(y, v', v", •••, v(n))=o, which is charac
terized by the absence of jt\
The transformation x =_v, y = y', reducing the differential equation
to one of order ;/ — i constitutes the usual method ior solving an
equation of this type. (El. Dif. Eq. § 58.)
IO2 TIIKOKY OF DIFFERENTIAL EQUATIONS §30
The remark of I', § 28 with reference to the introduction of
canonical or other variables when a group is known under which
a given differential equation is invariant applies equally well here.
I. The general tvpc of differential equation of tJie ntJi order
invariant under Uf = y— is f( x, --,"—, •••, •- - - ] = o, which is
'
y y
characterized by being homogeneous in y, v', v", ~-,y(n).
VI. The genera! type of differential equation of tJie ntJi order
<V , V • f(y i" r" v(H} \
invariant under C/ES.V/ -f- r\— - is J\ ~ , — ,, — — ;,> •••?" = o,
c).v " By \.v'' .v'' .vr xr )
which is characterized by being homogeneous in x, y, y',y", -',y("\
when these elements are given the weights i, r, r—i, r—2, ••«,
r— n respectively.
As special cases of this group may be mentioned
IV: r=i,
V: r=-i, /(AT, .vV, W, -., .v"* V"0= o,
III' : r=o, /( y, xy\ ^;r", ••-, .v"_v'"))=o,
III : r=&2. The invariant equation in this case is more readily
recognized by the other characterization given under 111 above.
VII. The general type of differential equation of the ntli order
invariant uihler
or ^,» - <";> = .v, ^ - <£, ^,r - <^>, • •-, n~ - -?.
Note. — An interesting equation of this type is the complete linear
equation
(80) y> + I\y( " -1' + 7'8y »-2> -f ... '-h /^O'" + /^,-ui'' 4- /'«.v - A^.
j 30 niKKKKKXTIAL EQUATIONS OK MIOHKR ORDER IO3
If T= ]<o is a particular solution of the abridged equation obtained
by replacing X by zero, (80) is invariant under Uf=yn =- •
The transformation (79) y=y0y' —}'0'y (or y=yHy, resulting from
the introduction of canonical variables) reduces (80) to a linear equa
tion of order // — i (fi/. Dif. Eq. § 59), but the resemblance of the
resulting equation to the original one is not as striking as in the case
of the linear equation of the second order (VII, Note, § 28).
XII. 77ir gene i\i I type of differential equation of the nlJi order
invariant under rf=a^+l& is f(bx - ay, v', r", •••, vril))=o.
d.\- dy
CHAPTER V
LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF THE
FIRST ORDER
31. Complete System.*
THEOREM I. — If <£(.v, v, z) is a solution of the two independent 'f
linear homogeneous equations
=Pi(x, y, z)
B/U*,* z) +&(*,* z) +*,{*>* z} = 0,
it is a/so a solution of
4- (A, Q, -
where (A^A^) is the alternant of the operators A^ and A., (§ 14).
For (AiA2)<f> = A^A^) — A.2(Al(f>) = o, since A^ =o and . /L,<^ = o.
* Only so much of the theory of complete systems and only such methods for their
solution as seem necessary lor our immediate purpose are given here, tor an excel
lent detailed treatment of this subject the student is referred to Goursat-Bourlet, Inte
gration i/t's equations au\ derive es partielles du premier ordre.
t r equations of this type in n variables are said to be independent if it is impossible
to find r functions a\, ff^>, •••, <rr of the variables such that
In the case of r = 2, this amounts to saving that the equations are independent if
one of them is not a multiple of the other.
104
§3i LINKAR I'AKTIAL I >I !• !• I-.UKNTI AL EQUATIONS 105
If three linear equations in three variables A}f= o, A.j,f = o,
A»f=o have a common solution <f>(x, y, z] other than a constant,
dd> dd> dd>
_r satisfy the three homogeneous linear relations
dx dy dz
Since <f>(x, y, z) is not a constant, -, - , cannot all be
identically zero. Hence
i a *i
a ^i
It follows that three functions <ri(x,y, z), <r.2(x, y, z\ (rs(x, y, z) can
be found * such that
(8 1 ) o-i^i/4- <r2^2/ -f- vsAJ = o ;
i.e. the three equations are not independent. Hence follows
THEOREM II. If the three equations in three variables A\f= o,
A.>f= o, ^/;i/= o have a common solution, other tlian a constant,
they are not independent; or stated otherwise, three independent
linear homogeneous partial differential equations in tJiree variables
cannot liave a common solution, other tJian a constant.
From Theorems I and II, it follows at once that if A^f= o and
A.,f= o have a common solution,
(82) (^M,)/EE Pl(x, yt ZJAJ+ p,(.v, r, • ,. /._ /;
* Thus, for example, one may take for <r\. o-._>, <T;} any tliree functions proportional
to the col.u-tors of the i-oricsMoudini; dements ot any column m A.
106 THEORY OF DIFFERENTIAL EQUATIONS §31
Clebsch gave the name of complete system to a pair of independent
equations A^f — o and A.2f= o, which are connected by the rela
tion (82). The last statement may therefore be put into the form
THEOREM III. If Alf=o and A f •=. o have a common solution,
they form a complete system.
Conversely, we shall prove the very important
THEOREM IV. If ^/= o and A.1f = o form a complete system,
they have a common solution.
In order to do this it is necessary to prove two lemmas.
LEMMA I. If A^f — o and A.,f = o form a complete system, any
pair of equations formed of independent linear combinations of these,
also form a complete system.
The equations
(8 )
= 2, y, z) + x, y, z).>= o
are independent if \^2 — ^i ^ °- Then y^/and A.J can be found
as linear functions of A^f and >l>/from (83).
Since A-^J '= o and A<2f— o are supposed to form a complete system,
is seen to be a linear function of A^f and A*f, and therefore of
>f,/and A-if, which proves the lemma.
Moreover, any common solution of sl}f= o and A.tf= o must be
such for >0j/==o and yf^/^o, and vice versa. Hence the two systems
are said to be equivalent, or each is said to be equivalent to the other.
A system equivalent to the original system is obtained if the equa
tions of the latter are solved for two of the three partial derivatives
-^-, / , ; • This can always be done, since all three of the deter-
-
minants in the matrix
y>
Q,
§3i LINEAR PARTIAL DIFFERENTIAL EQUATIONS IO/
do not vanish identically, AJ= o and A.,f= o being independent
equations. If, in particular J\Q, - 1>,Q, = D =£ o, the equations
may be solved for -j- and ^ , thus giving
where Rl=. *, = =:. Here *,=-§,
/xj = - ^ , X, = - ^ , n»=-\ and X^o - X,/^ = -^ =£ o, since all
functions involved are supposed to be generally analytic. Hence
equations (83') are independent. This fact is also obvious upon
inspection, since the first equation is free of v~, while the second
/)/
does not contain ~ • Moreover
dx
(84) (A^f^o.
For, since AI/= o and A>if— o form a complete system
(82) . (Aj.yEEp
In the case of equations (83')
which is free of both /- and - - . Hence pi and p., in (82) must both
dx dy
be zero, and the form (84) follows.
A complete system for which pl = p2 = o is called a Jacobian* com
plete system. We have thus established
* Originally this term was applied only to a complete system in the special form
(83'). Lie and other mathematicians, however, used it, as above, to apply to the more
general class of complete systems; (SIT Lie, DifertntialgUifktaig€*t p. 202; Goursat-
BourU-t, /<>6-. cit., p. 347; also Encykkptidie dcr M^t/icinatisi/icn U'lssi-HSi/iaftctt, Hanil
HI. P- 315).
108 THEORY OF DIFFERENTIAL EQUATIONS §31
LEMMA II. — A Jacobian Complete svstein can always be found
equivalent /<; a gircu complete s\ stein.
Remark. — It should be noted that this equivalent Jacobian sys
tem is not unique, since starting with such a one, the system obtained
by taking any pair of independent linear combinations of these equa
tions with constant multipliers is another system of the same sort.
It is easy to show that a Jacobian complete system has a solution.
Suppose that A-^f ' = o and A»/= o form such a system. Then
(84) (*A)/= Ai(A'./)-A*(Ai/)= o.
If u(x, y, z) and v(x, r, 2) are two independent solutions of one of
the equations, say Alf=o, any function of // and v will equally well
satisfy this equation. \\. remains to find such a function of them,
F(tt, v), that it shall also be a solution of the other equation >f2/— o ;
that is,
(85) . A*F(u, v)s °f A,u + ~A,v = o.
()// VV
Replacing / in (84) by u and v successively,
= o and A^A.v— AJiA^= o.
Since A^u = o and A\v = o, it follows that
Al(A.>!t)=o and Ai(A%v)=Q.
Hence Azu and Av are functions of // and v, say $(//, v) and ^(//, i>)
respectively, and the equation (85) to determine F(u, v) is
(85') <K", 7.')^+ t(»,V)d£=0.
The solution of this equation (which is known to exist by the gen
eral existence theorem) is ;i solution of the Jacobian system A}/=o,
A-J= o, and consequently of the equivalent complete system At/= o
and A^f= o. Theorem IV is thus proved.
LINEAR PARTIAL DIFFERENTIAL EQUATIONS
109
All that has gone before can be extended at once to homogeneous linear equa
tions in // variables.
Without changing a word in the proof of Theorem I we have: If
0 (•*'!> -*"2> •••» •*«) -is a solution of the two equations
+ /'•>„ (>i, .*••>, •••, .vw) ,f - = o,
<•/»«
zY is also a solution of (A\A>>}f = o.
As before, ?/" w equations hare a common solution, other than a constant, the
equations cannot t>e independent. For the determinant of the coefficients
must vanish. Hence a relation of the form
0V/ 1/ + «r.j//2/+ •••
must exist.
Starting with /• independent eqtUttioiU
.-/i/=o, A.2/=o,
Arf=o(2<r<n)
with a common solution, all the equations
(/MJ/zrO, (*V«= I, 2, 3, ...,r),
will also have this solution. Some or oil of th.-se equations may be independent
of the original equations. Adjoining these to the latter, the proems may In-
repeated as long as independent equation! can be lound. This process must
come to an t nd before the total number of equations reaches //. For it ha-; ju-t
been seen that there cannot be // independent linear equations in ;/ variables
1 10 THEORY OF DIFFERENTIAL EQUATION'S §31
having a common solution, other than a constant. We have thus obtained a
system of s equations
Aif=o, A-2/=o, -•-, A.f— o (r < s < ;/)
such that (AtAlty=piAi/+ p-j.-/,>/+ ... + p,A,f,
(/, K=l,2, 3, .-.,.*),
Such a system constitutes a complete system. We have thus shown that if ' r
equations hare a common solution, everv member of the complete system determined
by them has that solution.
It will be left as an exercise for the student to show that starting with any
complete system an equivalent Jacobian system* can be found. The method is
identical with that given above for three variables.
That a Jacobian complete system (and, therefore, any complete system) of s
equations in ;/ variables has n — s independent solutions may be proved in a
manner entirely analogous to that used above for s = 2, n = 3. To illustrate, the
case for s — 3, n = 5 will be given without detail :
The equation
has four independent solutions m, #•_>, «;, «t (F<t- Dif. Rq. § 79). The problem
is now to show that some function J'\u\, //-j, w;5, //4) of these will satisfy both
Arf — o and A%f = o.
Since ut for i — I, 2, 3, 4 satisfies A\f — o, it follows on replacing / by ;/, in
the identity
that A»iii is also a solution of A\f— o. Hence A»Ui must be some function of
«i» "2, »3f «4, say 0<(«i, //_•, //;•„ //.,), for / = i, 2, 3, 4. If /'Ms any solution of the
equation involving the four variables MJ, //._., n-\t u*,
it will be a solution of A\f— o and ./•_>/= o.
* A Jacobian complete system of J equations is one for which
(,/,./J/ :0(*,*-I,a,3,...,x).
See prt^vious footnote.
§§.31,32 LINEAR 1'AKTIAL DIFFERENTIAL EQUATIONS III
This equation has three independent solutions v\, v^ v-^. Any funetion of these
will be a solution of A\f— o and A*/ — o ; and conversely, every solution com
mon to A\f—Q and A-^f—Q must be a funetion of 7'j, z*>, v^. It remains to
show that some function, 4>(z>i, z>2, ^3), of them will satisfy A^f— o.
As before, it follows on replacing/ by ?',- in the identities
that ^;;^j is a solution of both A\f— o and A-if — o. Hence .•/;{7't- must be some
function of z/[, ZA-, ?',;, say ^,(^1, ?/.-, ?';;), for * = I, 2, 3. The function 4> may then
be any solution of the equation
This is known to have two independent solutions. Each of these is the-.efore a
solution of the complete system, and there can be no others.
32. Method of Solution of Complete System. — To actually find
the solution common to the members of a complete system A^f= o
and ^-2/— ° ^ ^ not necessary to pass to an equivalent Jacobian
system. If it and v are two independent solutions of one of the
equations, A^f — o, it is known that some function F(u, v) is a solu
tion of the other ; i.e.
dF A/?
(85) A2F(u, v) = A.u-g + A& ~ = o.
or
(86) f
Knowing that some form of *F(tt, v) must satisfy this equation,
whence — - and - - are also functions of u and v, — ^ must be a
OK dv A«u
function of // and v.* Hence (86) may be written as an equation in
these two variables only, and the usual method of solution for such
an cc juation may then be followed.
* It should be noted that in this ease, unlike in the case of a Jacobian complete sys
tem, A.M and A.v need not be Junctions ot u and v, although they may be.
112 THEORY OF DIFFERENTIAL EQUATIONS §32
Since (_AiAJ)f=Aif, these form a complete system.
Here //=jr, v = z are solutions of A^f= o. Then A»t(=v — itt
A»v = z = v, and equation (85) may be used to determine F\ thus
rV'' , QF
v-±- +v-- -=o.
O// 027
The general solution of this is any function of - . Hence the com-
u
mon solution of the complete system is any function of -.
x v y
Or starting with u = -, v = " , the solutions of^.,/=o,and noting
y z
that A& = - , Af) = - , whence -^ =-^ = -, equation (86) is
>' 2 A& z u
dF , 7- 5^
-- 1 --- = o.
au 1 1 dv
Its solution is -=-v, giving the common solution of the system 01
equations.
Ex.2.
,~(x- + r +.v- + + r- «f-hjw -= o.
These form a complete system, since (AlA.?)/= A^f.
}. 2u = ^- Z-
* 2
-xz-x
» = -,#s- are solutions of A}/=o.
*
= -- = _ and equation (86) is
r 7^
Its solution is //-' -4- ir. Hence the common solution of the com
plete system is any function of - ~*- .
§§32.33 I.INKAK I'AKTIAI. I IIFFEKKNTIAI, EQUATIONS 113
Ex. 3. AJ-r - *= o, A,f= = o.
Ex.4. JJ-x + z
Ex.5. W*V + t£ + 'tf-i A^X!i{ +,f + sf=o.
ojc a dz ox d dz
Ex. 6. AJ= (A- -.v + 8) - 2j -f- (,v - y + f)= o,.
Ex. 7. ^/^ (» - j) 4- (yz - A-)+ (i - ^)= o,
- o.
33: Second Method of Solution. — If 4>(jc, _y, 2) is a solution of the
complete system Avf= o and ^./— o, the equations
give : : ='
Since the total differential equation which has c/>(.\, r
for solution is .
114 THEORY OF DIFFERENTIAL EQUATIONS §33
or differs from it by a factor involving ,r, y, z only, this equation
may take the form
(87) «?A- Q^dx+(R&-Rfddy+(P^-PiQdti=**
The problem of solving a complete system is thus reduced to that
of solving a total differential equation (87). At times the actual
work involved in solving (87) turns out to be simpler than that re
quired by the method of the previous section.
Besides the usual methods for integrating total differential equations (see
F.L Dif. Eq. Chapter VI) the following method due to Dubois-Reymond may be
mentioned.
Instead of letting one of the variables, say c, be a constant temporarily, as is
usually done, let it be a linear function of the other two, thus
z — x-^ay
where a is an arbitrary constant. This relation carries with it
dz — dx -\- ady.
Eliminating z and dz from these two and the total differential equation, there re
sults an ordinary differential equation
Jlf(xt >', a)itx + A\x, y, a}dy = o f
whose solution \}/(x, y, a} = const.
* Equation (87) may be put in the convenient determinant form
dx dy dz
(?i A'j
Q-i RI
which expresses the condition that the above three homogeneous linear equations in
^, 2*. ^ are consistent.
dx dy dz
t If it happens th.it this differential equation does not contain a, some other linear
relation among the three variables containing an arbitrary constant should be tried
leading to a differential equation in two of the variables only and containing the arbi
trary constant.
§§33,34 LINF.Ak PARTIAL DIFFERENTIAL KOUATIONS
gives, on replacing a by its value in terms of x, y, z,
= const.,
which is the solution of the total differential equation.
This method requires the solution of only one ordinary differential equation
instead of two, as in the usual method, when an integrating factor is not known.
But in actual practice, this theoretically simpler method may not prove as de
sirable as the other.
Ex. The examples of § 32 should be solved by the methods of
this section.
Thus for Kx. I the total differential equation to be solved is
dx d dz
o o
x y
Its solution is -^ = const.
= z dy - ydz = o.
For Ex. 2
dx dy dz
x y z
x- + y1 + yz x2 -f y2 — xz z(.r + y
= 0
becomes, on multiplying the second row by x + y and subtracting from the third
row,
dx dy dz
x y c =0
. y —x o
xz dx + yz dy — (x~ -f y2)dz = O.
An obvious integrating factor is
*(•**
, and the solution is -- ^ = const.
34. Linear Partial Differential Equation Invariant under a Group. —
The homogeneous linear partial differential equation of the first order,
(88)
Il6 THEORY OF DIFFERENTIAL EQUATIONS §34
has two independent solutions <l>i(x, y, z) and <f>o(x, y, z). Every
other solution is some function of these.
The result of transforming (88) by the transformation
( 89) xj_ — <f> (x, y, z), }\ = \j/(x, y, z), zv = ^(x, y, z)
is ([15], § n) the new equation
(90) ^+^+^1=0;
wliere A<j>, A\l/, A% are to be expressed in terms of xlt ylf %. If (90)
is the same equation in the new variables as (88) is in the old ones,
or differs from it by a factor, the transformation (89) is said to leave
the differential equation (88) unaltered. In this case it must trans
form both <#>! and <£2 into solutions again ; that is, they are either left
unaltered by (89) or they are transformed into some functions of
themselves by it.
Let us find under what condition (88) is left unaltered by every
transformation of the group
We have seen ([7], § n),
«kC*i, v,, *,) = *,.(.v, y, z) + U^ '- + LT^ £ -f .-
In order that this be a function of ^Lv, v. : ) and <^L,('.v, r, z) for all
transformations of the group, i.e. for all values of /, it is necessary that
tf<fc=/X<h, «fe)f (1=1, 2).
It is readil seen that this is also a sufficient condition. For
67W,, ^ = Ufr -h u^ = WF} + . p^
LINEAR PARTIAL DIFFERENTIAL EQUATIONS
117
which is again a function of fa and fa. In the same way it can be
shown that if (Jkcf>i is a function of fa and fa, (7k+l<f>i is. Hence,
(91) is the necessary and sufficient condition that the equation*
whose solutions are fa and fa shall be invariant under the group.
It is desirable to have a condition expressed in terms of the differ
ential equation itself. The linear equation
(92)
has fa and fa for solutions when A/= o is invariant under Uf. For
(UA)fa = L/Afa - AC/fa = U(o) - AF^fa, fa) = o
(1=1,2).
Since (88) and (92) have the same solutions, they must be the
same equation, to within a possible factor, by the previous footnote.
* A unique linear differential equation of the form (88) (to within a possible factor
involving the variables only) is determined by two independent solutions. For if
0! and <J>.2 are the solutions of
4fte/»£+.0if +*£-*
dx dy -dz
then
whence
. p _
d.y
So that the differential equation having
convenient form
= -= -- r .--=— = ,
dx By 62
- 0^0-2 i ^ ^0-2 i r> dfa
=P-= — H Q -^ - + R -7; — = o,
dx dy dz
' f^2 • ^1 dfa __ 501 d<t>-i . 50i 50-_> _ (901 502
s dy d* dx dx dz ' 3-»- (> 3r ^-^
and <.> for solutions ma be written in the
dx dy d*
50i 301 a^
3^ 3;-
502 50-2 30-
dxt dy d=
Il8 THEORY OF DIFFERENTIAL EQUATIONS §34
Hence when Af— o is invariant under Uf
[31] (UA)f=\(X,y,z)Af.
Conversely, when [31] holds, (88) and (92)* have the same solu-
tions; then = -o, (1 = 1, 2).
Since A<^i = o) it follows that A (Ufa) — o ; hence C/fa is a solution of
(88), and must be a function of fa and fa.
Therefore [31] is both the necessary and sufficient condition that
Uf leave Af = o unaltered.^
Thus, the group Uf=x & +y |f + z & leaves J/&&+&+ |/= o unal
tered, since (U^)f~- (§1 + |/ + |A =_ ^
Vd* 6>>' 6s /
Similarly the same group leaves Af=y J- — x~ + z^=o unaltered, since
It also follows from this that the group Uf=y |C - * ?/ + 2 Cleaves the
5/" 5/ 3/ ^ ^^
equation ^4/= x -^- + y ~ -\- zjt = o unaltered.
Remark. — From the form of the condition [31] it is obvious that
if an equation Af = o is invariant under each of a number of groups
£/,/, £/>/, •••, Urft it is invariant under Uf=alUlf+a,U,f+ •••
-f arUrf, where alt a.,, •••, ar are any constants.
* If \(x, y, z) is identically zero, in other words if (L?A)/=o for all functions f,
[31] is still a sufficient condition that Uf leave Af—o unaltered. In this case one
cannot speak of the equation (92); but writing the identity ((J.-f)/~~o in the form
UAf^AUf, it follows that AU^ = Q since UAQi : = U(o) = o. Hence U^ is a
function 0i and <t>-> as above.
t Using the method of the previous footnote, it can be shown that a homogeneous
limMr equation in n variables is determined, to within a factor by its n — i independent
solutions. The argument of this section therefore applies without change to such an
equation. Hence [31] is the condition that Af— o, involving n i\viat<les, shall be
invariant under the group Uf £i /' I & „ + '" ' ^. ^ ' In ? X5 essentially the
Qx\ (j/.vo ('!»,,
same method for the case of two variables was carried out.
§§34,35 LINEAR PARTIAL DIFFERENTIAL EQUATIONS I 19
Ex. Determine which of the equations below are left unaltered
by each of the following groups :
dx
2. iy=-*
ox dy
3. V/=*
dx
4.
<1f ., df . df
d.Af=vfx-rfy+(*+flj=<>.
35. Method of Solution of Linear Partial Differential Equation In
variant under a Group. — If the equation Af= o is invariant under
Vf
[31] (UA\f=\Af,
i.(\ IJ— o and Af=o form a complete system. Hence the methods
of §§ 32 and 33 are available for finding one of the solutions of Af— o.
* While lr/=P(*ty, c). //"leaves ///"— o unaltered for all forms of p(r,y, c), such
a group is said to lie //•/:•/,// because it is «»t no service in solving Af— o. We shall
presuppose that tin- group //under consideration here is not trivial.
120 THEORY OF DIFFERENT] AL EQUATIONS §35
Having thus found $(x, y, z), a common solution of 6^=0 and
Af=o, a second solution of A/=o may be found in the followin'g way :
Since <(>(x, y, z) is not a constant, it must involve at least one of
the variables, say z. Replacing z by the new variable
z=<l>(x,}', z),
the equation and the group take the forms ([15], § n),
Af= P(x} y, z}^ + Q(x, y, *')|^= o,
ox dy
Uf = |(>, yt z)
since A$ = o and U$ = o.
Here P, Q, |, T\ are what P, Q, £, -q respectively become when in
them z is replaced by its value in terms of x, y, z obtained from
z = <j>(x,y, z). Here z plays the role of a constant since the coeffi
cients of — in A /and Uf are both zero. To solve Af=o we pro-
dz
ceed to the corresponding ordinary differential equation
Qdx — Pdy = o.
This is invariant under Uf. Hence the methods of §§ 12 and 20
may be employed.
Remark. — When the usual Lagrange method (see RL Dif. Eq.
•§ 79) is practicable, it will, as a rule, prove simpler than the method
of this section. As an exercise it may be desirable to solve the
examples below by both methods. But the Lie method is of inter
est theoretically and may prove valuable when the other method can
not be carried out.
Ex. 1. Af~ 2 xv- - 2 *y+(/- x'*)z= o.
dx ' (>v dz
The coefficients are homogeneous and of the same degree. Hence
this equation is left unaltered by the group Uf '= x -;'- -f v -£- + *-£• ',
i).\- ~ dy dz
as a matter of fact, (L7A)f = 2 Af.
§§35. 36 LINEAR PARTIAL DIFFERENTIAL EQUATIONS 121
xv
By the method of § 32 or that of § 33, -^ is readily found to be
z~
the common solution of Af = o and Uf= o.
The transformation z = -+- reduces A/=o and Uf to
Af = 2 xy- -J-— 2 x2y ^~ = o and Uf = x-^-+y-^-
dx dy dx ' dy
respectively. The corresponding ordinary differential equation is
2 x2y dx H- 2 xyz dy = o.
Lie's integrating factor -?- (or the obvious integrating
2xy(x*+y*) \
factor -- ] leads at once to the solution x2 -f-j^ = const. Hence two
independent solutions of Af= o are 5| and
z~
dx / dy ' ' dz
Ex.3. ^=^ + v
oy
Invariant under Uf = (x +}')-- +(^ + v)— -f 2 z^-, as well as
dy dz
*y\
Ex.4. Af = (xz-ti& .
dy
[Invariant under Uf=x+- + v— •
dx dy_\
36. Jacobi's Identity. — For further development of the theory it
will be necessary to have available a certain identity first noted by
Jacobi and known by his name :
If Aif, A^fj A^f arc three homogeneous linear partial differential
expressions in any number oj
(93)
122 THEORY OF DIFFERENTIAL EQUATIONS §§36,37
This may be verified directly in the case of three special forms,
and also in the general case for two variables. This is suggested as
an exercise to the student.
Probably the simplest way to prove the theorem is the following,
due to Engel :
Since (A^A^f == A^f- A,AJ,
((A,A,)A,}f~ A,A2A,f- A.A.A.f- A.A^f + A^A.f,
A.A.A,/,
The sum of these is obviously identically zero. Hence the identity
(93) is established.
37. Linear Partial Differential Equation Invariant under Two
Groups. — If the equation Af= o is invariant under two distinct*
groups UYf and
[31']
Jacobi's identity (93) for UJ, U^f, A/ is
Using [31'] and obvious properties of alternants (§ 14), this
bec°mes
where /x = Uv\^ — U<X\. Hence the
THEOREM. — If Af = o is invariant under U\f and £/,/, // is also
invariant under (£/, 6o)/.f
* Two groups £/!/ and U^f are said to be distinct with respect to the equation
/(/— o, provided no relation of the form
(94)
exists, where al and a.2 an- constants and p is any function of the variables. For it is
obvious that if ( \ f leaves . // o unaltered, U.,fi-icU\f-\- pAf will also do so for all
choices of the constant c and of the function p(x,y, z).
f This theorem holds, and is proved in exactly the same way, for n variables.
37
LINEAR PARTIAL DIFFERENTIAL EQUATIONS
123
If (£/;£/,)/ is not of the form
(95) *iUJ + aMf + P(x, y, z)Af,
where a± and a2 are any constants and p is any function of the vari
ables, it is said to be distinct from U\f and U^f with respect to the
equation Af = o. In this case the theorem gives a new group under
which the equation is invariant. The theorem may then be applied
to this new group and one of the original ones. And so on.
Remark. -— It is important to note that there always exists a linear
relation between four homogeneous linear partial differential expres
sions of the first order in three variables.* For eliminating -£. *-
dx dy
-•- from the four identities
*
= ft -7-
ox
- - : ,
dy dz
'
dz
the linear relation
U,f ^ r,,
ft 17,
is obtained. In general the coefficients are functions of the variables.
* Similarly, there is always ;i linear n^laticn between n -\- 1 sueli expressions in n
variables.
124 THEORY OF DIFFERENTIAL EQUATIONS §§37,38
As a consequence we always have
If it turns out that «j and «2 are constants, this is of the form (95), in
which case (£/i&Q/is not distinct from £^/and
Thus the equation Af&& + & + &= o is left unaltered by
d* dy dz
dy dx dy
since (£/i/l)/=o, (U*A}f=- 2 x Af.
Moreover (U^U^f=-(x - y~] ^ 2(y - z)(x -
dy z
also leaves Af= o unaltered, since ((U\U*)A)f=o. It is readily seen that
Again
' 4- 2(7 -«)(*- 2
x—y \x-y
also leaves Af — o unaltered, as is readily verified. And so on.
38. Methods of Solution of Linear Partial Differential Equation
Invariant under Two Distinct Groups. — Two important cases are to
be distinguished :
A*. If a relation of the form
(9 7) UJ= u(x, y, z) UJ- + p(x, y, z)Af
§38 LINEAR PARTIAL DIFFERENTIAL EQUATIONS 125
exists,* where a is not a constant, 6^/is still considered distinct from
Uif. In this case a(x, y, z) is a solution of Af= o. For, since
Af= o is invariant under U^f, (£72^)/must be a multiple of Af. But
- AP Af
= (a\l-AP)A/-Aa UJ.
Since U\f is supposed to be not trivial, i.e. not a multiple of Af
(§ 35 )> the onty waY in wnicri (£4<4)./can be a multiple of Af is by
having Aa = o. Hence « is one of the two independent solutions
of Af= o to be found, f
To find a second solution of Af= o, several possibilities may arise
which will be mentioned in the order of desirability :
i° Since A/=o is invariant under U^f, 67i« is also a solution of
Af=o [(91), § 34]. If U\<t turns out to be distinct from a, it may
be taken as the second solution necessary to give the general solu
tion of Af— o.
2° If U\u,\ is a function of a or a constant other than zero, two
methods are possible :
* A linear relation between Af, U±f, C/2/will show itself by the vanishing of the
determinant of their coefficients, thus
A =
P Q R
~io.
»?2
Here
t Conversely, if a is a solution of A/=o and f/i/is a group that leaves the equation
unaltered,
[35] U
will also leave it unaltered no matter what be the form of p(x, y, z). For
( U2A)/= (a f/i + pA, A)/= (a\i - Ap)Af.
since Aa=o. [Compare (35) § 17.]
I Since (S.2 a = a U± a -f- pAa. = a U\. a, it is sufficient to consider U{a only.
126 THEORY OF DIFFERENTIAL EQUATIONS §38
(a) The solution common to^//=:o and £/i/~=o (or £72./= o)
may be found by either the method of § 32 or that of § 33. Since
£/,« ^= o, this common solution will be independent of a.
(fr) Since a must contain at least one of the variables, say z, the
introduction of the new variable z = a (x, y, z) in place of z reduces
A/ = o to one in two variables,
z appearing as a constant since the coefficient of -*- is zero. (Com-
dz
pare § 35.) But since L\u=f= o, the above equation must be inte
grated, without any further assistance from the groups U\f and U.2f.
3°. If £/i« = o, the method of § 35 is available ; thus the intro
duction of the new variable z gives the same differential equation as
above, but now the transformed group
under which it is invariant also leaves z unaltered. Hence the
methods of §§ 12 and 20 are available for solving the corresponding
ordinary differential equation
B. If no relation of the type (97) exists between Aft U±ft U>>f,
the relation
(96) (UiU*)f= a,(x,y, z)l\f+ «,(*,;•, *) *V+ P (*,;•, z}Af,
which always exists (Remark § 37), will prove of service if «t and a.2
are not both constants ; for a^ and a., are solutions of Af— o, as may
be seen from the following consideration : *
* By exactly the same kind of 'reasoning as that employed here, the following gen
eral theorem can be established. (It is suggested that the student carry out the proof.)
Jfthe equation in n -variables
§38 LINEAR PARTIAL DIWKKKNTIAL EQUATIONS I2/
By the Theorem of § 37, ( £/i £/,)/ leaves Af= o unaltered. Hence
But
a,(U,A)f-
u,\., — Ap)A/— Auv L\f
Since no linear relation is supposed to exist between Af, U\f, U>>f,
the only way in which ((C/tCfyA)/ can be a multiple of Af is by
having Aa^ = o and Aa.2 — o. Hence rq and a.2 are solutions of
Af= o.*
i° If «, and a., are two independent functions of the variables, the
general solution of A/=o is known witliout any further work.
2° If one of them, say a , is a function of the variables, while the
other, a.,, is either a function of av or a constant, use may be made of
the fact that £/i«i and 6£«j are also solutions of A/=o [(91), § 34].
If either of these turns out to be a function distinct from alt it i«ay
be used as the second solution.
/j invariant under r -f- I distinct groups U^f, U.±f, •••, Ur+\f, and if no linear relation
exists between .If and r of the lr/'st but
Ur+l/=
then MI, Ct-i, •••, ctr are solutions of Af= o.
* The student should have no difficulty in showing that, conversely, if «, and «o
are solutions of Af= o, and 6^/"and £/>/"are two groups that leave the equation unal
tered, the group
[35']
will also leave it unaltered no matter what be the form of p(v,y, z). (Compare [35]
above.)
128 THEORY OF DIFFERENTIAL EQUATIONS §38
3° If both £/!«! and £/>«i* are either functions of nv or con
stants, either of the methods (a) and (If) of A, 2° may be employed.
Or,
(a) if one of U&\ and £72«i is zero, the method of A, 3° is
available,
(b) if neither is zero, the group Vf= U^U^f— U^U^f leaves
Af= o unaltered, and Va.^ = o ; hence case (a) exists.
4° If both «t and «2 are constants, say a^ and a2, the solution com
mon to Af= o and U\f=- o, and that common to Af— o and
[/2f=o may be found by either of the methods of §§ 32 and 33.
Moreover, these solutions will be independent since there is no
linear relation connecting Af, L/\ft £/>/. (Theorem II, § 31.) We
shall show, by a method due to Lie, that an integrating factor for at
least one, and sometimes for tfoth, of the total differential equations
arising in the method of § 33 can be found in this case. (But it is
possible, at times, to find by inspection, an integrating factor that is
simpler than the one given by the following method) :
In ( Ui U.2 )/=&i U\f+a2 Uzf + p A/ either al and a2 are both zero
or they are not.
(a) If *! = a, = o, ( U, Uz}f= PAf.
Since Af= o is invariant under [/.2f,
If <£(je, j, z) is the common solution of Af= o and 6r1/= o,
= o, since ( U\ (72)<f> = U\ U<& — U^ U^ = pA<}> = o ;
= o, since ( U«A)$ = U»A<$> — A U<& = \2A<f> = o.
These identities can hold only provided £/2<£ is a solution of both
—Q and Af=o; i.e. U.& must be a function of <f>, say
* In this case ( U^U^d.^ will also be a function of U^ or a constant, including zero,
because of (96).
LINEAR PARTIAL DIFFERENTIAL EQUATIONS
129
Moreover jF(<£) ^ o, for, as noted above, Af= o, Uvf= o, U»f=- o
cannot have a common solution, since they are independent. As
was done in an analogous case in § 12, </>, the common solution of
Af~ o and L\/= o, may be chosen in such a form that £/><£ = i . It
must then satisfy the three equations
dx
dy
dy
a-r r)2-
dx dy
-*
dz
These equations determine -S -, - } whence <^> is obtained from
dx dy dz
<l<l>=*±dX + *+<ly+d-+t
dx dy dz
by the quadrature
\dx dy dz
P Q R
6 171 Ci
, where A =
£> !/•> £2
In exactly the same way, ^, that form of the common solution of
Af— o and L7.if=o for which C^ = — i, may be obtained by the
quadrature
The determinant A is thus seen to be an integrating factor for each
of the total differential equations arising in the method of § 33 for
finding the two independent solutions of Af= o.
130 THEORY OF DIFFERENTIAL EQUATIONS §38
(<£) If only one of a^ and a., is zero, let a2 = o. Then
In precisely the same way as before, tt is seen that, if <£ is the
common solution of Af = o and U-^f = o, U<$> = -/yX0) ^ °- Hence
that form of <£ for which U«$ ~ i is given by the (quadrature
dx dy dz
r Q R
& rj} &
To find a second solution of A/ = o, independent of <j>, either the
method of A, 3° may be employed, or the common solution of
Af = o and £/2/= o may be found by one of the methods of §§32
and 33.
(f) If both (7i and a2 are different from zero, consider the two
groups
These are obviously distinct and leave Af = o unaltered. More-
We are thus under case (/;) and the method for that case may be
employed.
Note. — For practical purposes it may be worth noting, that the
choice of the groups U^f and Vf = a^ U^f + ti.2 U^f also leads to
case (f).
Remark i. — A hasty survey of the processes involved in the
methods to be employed in the various cases considered in this sec
tion, brings out the fact that when two distinct groups are known
under which the equation Af = o is invariant, the solution of the
latter can be obtained by means of quadratures only, except in the case
of A, 2°, where one ordinary differential equation of the first order
§38 LINEAR PARTIAL DIFFERENTIAL EQUATIONS 131
must be solved. In certain cases, such as A, i° and B, i° and 2°, no
integration whatever is required. In the above scheme, certain alter
native methods involving the solution of differential equations have
also been suggested, for in certain cases these processes may prove
simpler of execution than those involved in carrying out quadratures.
Remark 2. — It is easy to prove the existence of a pair of groups
U^f and U^f under which A/=o is invariant, and for which no
linear relation of the form (97) holds. For Af = o has two indepen
dent solutions (j)l and <£2. These are independent with respect to
at least two of the variables, say x and y. Introducing the new
variables . /
x = $!(#, yy «), y = £(«, j, 2), z = z,
Af = o takes the form
= 0.
dz
By inspection C/i/= -*- and U.,f = -^- are seen to leave the differ
ed dy
ential equation unaltered. Moreover there is obviously no linear
relation between -J-, SL, §£• Passing back to the original varia-
dx dy dz
bles, Af=o will be invariant under the groups (/if and U»f into which
U}/ and Z7,/ are transformed, and no linear relation can exist now.
Ex. 1. ^/ = (.v + v., v"-.r 2S/=
This equation is invariant under
U,f = ,(.v + ,,f +,(* + j02+ [(.vv + 2 ^)(.v + v) + 4 .vvs]
as may be verified easily.
132 THEORY OF DIFFERENTIAL EQUATIONS §38
Here A = o, and U*f = (yz + zx + xy) UJ - xyAf, (A).
.\yz-\- zx -f xy is a solution of Af = o.
Moreover U^yz + zx + xy) = $(yz + zx + xy) + (x— y)2 is also a
solution (A, i°).
Taking account of the first solution, the second one may be
replaced by x — y. Hence the general solution of Af= o is
<&(yz + zx -h xyt x — y) = o.
Ex. 2. Af^xz-y^ + ^z-x^ + ^-z^-^v.
ox oy oz
This equation is invariant under
9f ' df
=x-£-+yJ- and
dx dy
A = o, and £/>/ = (,
.*. ^ -f- yz is a solution of Af •=• o.
Moreover ^(jc +yz) — x +yz ^ o (A, 2°).
To find the solution common to Af=o and [\/=o the method
of § 33 requires the solution of the total differential equation
y(i — z~)dx — x(i— v)dy +(yi — x*)dz = o.
An obvious integrating factor is - - - , leading to the
(f — *
solution
-
\x-yi ~z
x -\-yz- (y-\-xz)
The left-hand member of this is, therefore, a second solution
§38 LINEAR PARTIAL DIFFERENTIAL EQUATIONS 133
Taking account of the first solution, the second one may be
replaced by y + zx. Hence the general solution of Af=o is
$>(x +yz, y + zx) = o.
Ex 3. A=x
ox oy
OZ
Ex. 4. A=
dx dy dz
E,5. Af=f + !f + ¥ = 0.
ox o oz
EX. 6. Af= (xz - y) + (yz - X) + ( I - Z*) = O.
dx dy dz
CHAPTER VI
ORDINARY DIFFERENTIAL EQUATIONS OF THE SECOND
ORDER
39. Differential Equation of the Second Order Invariant under a
Group. — The differential equation of the second order
(98) >'"=f(x,y,y')
is equivalent to the system of equations of the first order *
(99) *k = </v = __ * __
i / S(*,y,y')
If the solutions of the latter are
( i °o) // (.v, r, r ') = «, ?>O, j-, ]•') = /*,
the solution of (98) may be obtained by eliminating y' from the two
equations (loo).f
Instead of solving (99), one may find // and v as two independent
solutions of the corresponding linear partial differential equation J
The problem of solving (98) is thus reduced to that of finding two
independent solutions of (101).
If (98) is invariant under a group Uf, the equivalent system (99),
involving the three variables x,yt)\ is invariant under the extended
* El. /)>/, /-.-/. $ 68.
f Tin- c(iuations (too) ;in: two independent first integrals of (98). (Sec f 52,
Theorem IV.) t M- !>'/• l'-'l- k 79-
134
§39 ORDINARY, OK THE SECOND ORDER 135
group U'f. The effect of U'f on ?/ and r is, therefore, to trans
form them into some functions of themselves ; i.e. £/'// — <£(//, v\
f/'v = {f/(u, v). Hence the linear partial differential equation (101)
having u and v for solutions is also invariant under U'f(§ 34).
'Consequently the method of § 35 may be employed to find u and v.
Remark. — Since the invariance of (98) under Uf implies the
invariance of (101) under the extended group, and conversely, it
follows from the remark of § 34 that if (98) is invariant under each
of a number of groups U^f, U<>f, •••, Urft it is invariant, under the
group Uf= alUlf+ a.1U~.1f+ • •• -\- arUrf, where «,, a.,, •••, ar are
any constants.
This remark applies without modification to a differential equation
of any order, because the form of the condition [31], § 34 is inde
pendent of the number of variables appearing in the linear partial
differential equation Af= o.
EX. i. x
This equation is invariant under Uf= x -J- + ny -*- for any value
dx dy
of// (VI, § 28). In particular it is left unaltered by Uf^x^-^y^-.
d.v dy
Here Af^ % +y> gf+^- V) If = 0.
dx dy xy dy'
dx * dy
For the method of § 32* use may be made of the fact that
u = ^, v = v' are solutions of U'f= o.
.v
* The method of $ 33 requires the solution of
i/r </Y rtV'
X r— o, or (_y — .n'')( — l 4«* r +</v'j = o.
^ .v y I
y o
The evirlent integrating factor leads to the solution -vv avfj/.
/(>"•*•/) x
136 THEORY OF DIFFERENTIAL EQUATIONS §39
Au—xy'—y .^..-/P' — */) . Av _ XV9 V
All— - - - , AV — — — . . . — — — -- •-— = ---
x- xy An y u
Equation (86), § 32 is ^_^£^=0> ... p= uv = JZ'.
du u dv x
Introducing the new variable
4
j/ = ±]-, whence y = ^,
x y
x^—y- or xyy' —y2 is readily found to be the solution.
rvf
Eliminating y' from ^<- = a and xyy'—y- = b gives
i^
^jr2 — }r = ^
as the solution of the original equation.
Compare this method with that of § 27 or of § 28, I', Remark.
Ex. 2. n<"+/2=i.
Since x is absent, this equation is invariant under Uf=
(I', § 28). Here
By either of the methods of §§ 32 and 33, the solution common to
Af— o and (/'/= o is easily found to be/Vi —}'''•
Introducing the new variable
— y*, whence y =
§39,40 ORDINARY, OF THE SECOND ORDER 137
The corresponding ordinary differential equation
has for solution x — Vy2— y12 = const. Passing back to the originnl
variables this becomes x— yy' = const. Eliminating y' from
— y'2 = a and x— yy' = l>,
gives as the solution of the original equation
Ex. 3. y" =/2 + i. Ex. 4. x?yy" + (xy1 - yf = o.
Solve examples of § 28 by the method of this section.
40. Differential Equation of the Second Order Invariant under
Two Groups. — Since, if the two groups 6^/and U<J leave
(98) /' = ^(*,J,/)
unaltered,* the corresponding partial differential equation
is invariant under the extended groups 67iyand &>/, the methods of
§ 38 may be employed to solve (98).
Ex. 1. xyy" + xy'* -yy' = o (Ex. i, § 39).
This equation is invariant under U^f= x ~ +7^ an<^ ^/= y -•] "
,.
*Then (98) is also invariant under Uf^alUlf-{-a.1lr.1f (Remark, § 39). It is
possible that Uf may assume simpler forms than U^f or £/2/ for certain choices of
the constants a± and a.j.
138
THEORY OF DIFFERENTIAL EQUATIONS
A =
i y
x y
o y
xy
o
= 2 /-
Hence the method of B, 4, (a), § 38 applies. The solution com-
mon to Af— o and U-^f— o is
I
dx dy dy^
1 y
x y o
= r _£+*+*_ ,00*1
J ^ + V ^ V' ~ 8 ^
y y
1 A
The solution common to Af= o and £72/= o is
fo ^ dy
v, tfj-qh
J
= log
y'(y - x/ y'(y - *
The general solution of the original differential equation of the
yy'
second order is found by eliminating y from — = a and
- = b to be ax* — y2 = ct where c = ab.
' V
* The method of $ 32 is also available for finding these common solutions.
§40
ORDINARY, OF THE SECOND ORDER
139
Ex. 2. /' = F(x)y' + Q(x)y + X(x).
If y=\\ and_y=_)'a are particular solutions of the abridged linear
equation /' = iy + Qyt the general linear equation is invariant
under VJmy and VJ^y-d 28, VII, Note). Here
oy oy
4+
4+
ay
ay
where yj and j2' stand for -2-1 and -^2 respectively.
A =}'i}'2 —y<>y\ 3=- o, since the two particular solutions are sup
posed to be independent.
( C/i (/.>')/ = o. Hence method of B, 4°, (a), § 38 applies.
Since y = y\ and y =y2 are solutions of the abridged linear equa-
^°n> ,"_/5,'_|_/9; A ,H_p,l ft
whence P
Introducing these values in the expression for the solution com
mon to Af= o and Ul'/= o,
dx dy
i y jy-
o v,
we have
*»
140 THEORY OF DIFFERENTIAL EQUATIONS §40
Noting that -j'ij'0'2>i" -JiW) = -}'i"}'& +Ji>A', the first quad
rature is readily effected, thus giving
^ = ^y-j'i> _ c _M ___ ,/JC
y*y*-y*yi J y\y*-y*yi
Similarly, the solution common to Af •=• o and U»f= o is
^=^y-^v .. c v«* _^
- J -
The general solution of the original equation is found by eliminat
ing y' from </> = <r2 and \l/ = c\ to be
j = 'ij* + ^'2 -^i P— r^— , ^ + J2 f- -r^-7 ^«
^ j'U's - }'2}'i J )\yj -}',}','
Note. — It is an interesting fact that this form of the solution is
exactly that obtained by the method of variation of parameters
(El. Dif. Eq. § 49) from the complementary functionjy= clyl + c2y2,
as may be easily verified.
Ex. 3. y=7y + <2;<
This equation being homogeneous in y, y', y", it is invariant under
6i/"=7~-(IIIj § 28). Moreover, if y=yl(x) is a particular solution,
df
the equation is also invariant under (72f=yl^~ (§ 28, VIII, Note).
By
He,< ^.+
f/_ ., , ., >
=y\ ,~ -\-y\ -r-,-
d\- dy'
A =}'}',' -y')\ & o, ( 6V <72')/
Hence the method of B, 4°, (£), § 38 applies.
§40 ORDINARY, OF THK SECOND ORDKK 141
The solution common to Af=o and U»f=o is given by that
method in the form
AvV - PyJ - Q wWx-y,'<ty +yjy'
J **'->&
Replacing ^ Vj by its value y^' — P\y\, this quadrature is readily
effected, giving
A more convenient form for the solution is
To find a second solution of Af = o, introduce the new variable
r , i \I'dJC
j = *-*""(}>% -M'), whence /=^ +y6 -,
and / = o takes the form
dx y\ dy
The corresponding ordinary differential equation
(iy _ yj_ _ jfr^
dx r, " y\
is linear with the obvious integrating factor — • Its solution is
y\
v , c^1'1" *
' y \ — - ax = const.
}'i J y\
* Here y' appears as a constant, ($ 35).
142 THEORY OF DIFFERENTIAL EQUATIONS §§40,41
The general solution of the original differential equation is found
by eliminating y from
v Cf
<I> = a and *- — <$ I — <L\ = b
Cf
<$ I — <L\ =
J '
to be y = ay i I ' — dx -+-
= ".''. /%<*
?. — This is the same form of the solution as is given by the
usual method (EL Dif. Eg. § 53, i°).
Ex. 40y+y2=i (Ex. 2, §39).
This equation is also invariant under Uf = x ~ 4- y ; •
o.v o_y
Ex.5. y=y2+i.
Since jc and _y are both absent, two available groups are
Ex. 6. x?yy" — (xy* — y)~ = o (Invariant under VI for all values
of ni hence under III, III', IV, etc.).
EX. 7. (**+/)y'+ 2 O' -*)'')(! +y2) =0 (EX. 2, § 28).
Ex. 8. :rv" -f A:2j'/2 — 2 xy' + 2 = o.
This equation is invariant under I and III'.
41. Other Methods of Solution. — By making use of the properties
of what Lie calls r- parameter groups of infinitesimal transformations
(§ 43) the method of solving a differential equation of the second
order invariant under two groups can be modified so as to be con
siderably simpler both as to the number of cases to be distinguished
and as to the actual processes involved in obtaining the solution.
A brief study of these groups will be made in this chapter, leading
to the methods of solution in §§ 46 and 47.
§42 ORDINARY, OF THE SECOND oi$DER 143
42. Number of Linearly Independent Infinitesimal Transforma
tions, that Leave a Differential Equation of the Second Order Unaltered,
Limited. — Since a differential equation of the first order always has
an integrating factor, in general, (El. Dif. Eq. § 5) it is left unaltered
by an indefinite number of infinitesimal transformations, the general
expression for whose symbols involves two arbitrary functions (§ 15).
On the other hand, a differential equation of the second (or higher)
order is, in general, not left unaltered by any infinitesimal transforma
tion (see Note IV of the Appendix), although some of them are.
We shall prove the
THEOREM. — A differential equation of the second order cannot be
left unaltered />y more than eight linearly independent* infinitesimal
transformations*
Suppose that the equation
(98) ? = F(x,y,f)
is invariant under the nine linearly independent infinitesimal trans
formations {/if, U,f, ••', U»f, it is also invariant under
(102) Uf=
for all possible choices of the constants aif a2, •••, ay (Remark, § 39).
It is a well-known theorem in the Theory of Functions that, in
general, a unique integral curve of a differential equation of the sec
ond order and first degree (98) passes through two points, lying
within a definite region determined by (98). Suppose that /*,, /?_,,
/?!, /4 in Fig. 5 are four points such that each of the six pairs that
can be formed of them determines a distinct integral curve of (98).
The nine constants ah a*, •••, a9 can be so chosen that (102) leaves
* A set of infinitesimal transformations f/j f, lr%f, •••, Urfis said to be linearly in
dependent if there is no linear relation, with constant coefficients, connecting their
symbols; i.e. if it is impossible to find a set of constants <,. <•._,, •••, cr such that
144
THEORY OF DIFFERENTIAL EQUATIONS
§42
each of these four points unaltered. For, if their coordinates are
respectively (xlt }\), (x2,y2), (x3, ys), (x4, y4), the requirement for this
is the simultaneous satisfaction of the eight equations
(/= i, 2? 3> 4-)
These equations determine finite values of the ratios of eight of the
s to the ninth one (excepting possibly for peculiar choices of the
four points, which can be avoided)
because of the linear independence of
the nine transformations.
With the tf's thus chosen, the trans
formation (102) leaves the four points
Plt P2, Ps, ^4 unaltered and, therefore,
also the integral curves determined by
any two of the points, since integral
curves are transformed into integral
curves by a transformation which leaves
FIG. 5
a differential equation unaltered, and the four points were so chosen
that through any two of them passes a unique integral curve. Thus
through each one of the points, e.g. through /*„ pass three of these
invariant integral curves. The point Pl on these being left unaltered
by (102), their slopes at this point, which may be designated by j12',
.i'i:;'» Jut respectively, are also left unaltered by it. Hence if rf is the
coefficient of -- in the extended transformation corresponding to
d/
(102), it follows that
for x = xlt y=yi, y'=y\*,yv!, yu> Letting a, b, c be the values
§42 ORDINARY, OF THE SECOND ORDER 145
of n~> T*-^> ~^ respectively when x = xl9 y=y\, (103) gives
ox oy ox oy
the three relations a + ^, + ^,2 = O}
a + tVu' + OW2 = o.
Since the determinant of the coefficients
I >'13 J'lS
i ;•»' V2
is different from zero, a = b = c = o. Hence if •= o for every integral
curve through /\, whence every integral curve through Pl is invariant*
under (102).
In exactly the same way it can be shown that every integral curve
through each of the other points PZt P3, P+ is left unaltered by
(102).
If P is any fifth point in the region containing Plt P2, P3, P4, it
will lie upon at least two f integral curves each of which passes
through one of those points. These integral curves being invariant,
the point P is left unaltered by (102). In this way every point
of the plane (with, perhaps, exception of certain points determined
by the differential equation) is found to be left unaltered by (102).
The latter must therefore be identically zero ; i.e.
<*i UJ+ a, (72f+ • • • + a9 UJ = o.
* This follows from the fact that a unique integral curve of a differential equation
of the second order is, in general, determinol by tin- conditions th.it it pass through
a given point (x,y) and have a given slope y' at that point.
f If P does not lie upon any of the six integral curves determined by the four points
(which is the general case), this number is four ; it is three if P is on one of these
curves, and two if it is at the intersection of two of them.
146 THEORY OF DIFFERENTIAL EQUATIONS §§42,43
Hence any nine infinitesimal transformations which leave a differen
tial equation of the second order unaltered cannot be linearly inde
pendent. This proves the theorem.
The differential equation^1" = o is a simple example of an equation
that is left unaltered by the maximum number of infinitesimal trans
formations. For, since its integral curves are the straight lines of
the plane, y = ax + b, it is left unaltered by every projective trans
formation a4x + a:>
a*y + 0 9 a7x -f- a^y
In Note VI of the Appendix it will be seen that there are eight
linearly independent infinitesimal projective transformations.
Remark. — In the case of a differential equation of higher order
than the second, the following theorem holds : A differential equation
of the nth order (n > 2) cannot be invariant under more than n -f 4
linearly independent infinitesimal transformations. A proof of this
theorem maybe found in Lie, ContinuUrliche Gruppcn, pp. 296-298.
As in the case where ;/ = 2, a differential equation of order // > 2
is in general not left unaltered by any infinitesimal transformation.
On the other hand the differential equation y(n) = o, « > 2 is
invariant under each of the n -f- 4 transformations (Examples, § 26)
df df df df df odf ,df ,d/ , / N df
/ , ,;, *¥->yt-> *1L> *~ : {-> '">* T" ' ^/ + ^-1)^^--
dx dy dx dy dy dy dy dx dy
43. r-parameter Group of Infinitesimal Transformations. — Start
ing with a set of infinitesimal transformations U^, U*f, •••, Urft the
infinitesimal transformations, whose symbols are obtained from these
by applying the alternating process to them in pairs, may or may not
be linearly independent of them.
Thus, if
the transformations (f/i^/,.)/=, ( Ui
arc all indcjicndcnt uf them.
§43 ORDINARY, OF Till'; SFX'OND ORDKR 147
On the other hand, if
U\ f= x -~ , £/•>/= x „ , U?tf^.(x + y} -~- ,
u* uy o*
the transformation
( [/•> U-£)f=. x -• f~ — (jr 4- y ) jr~ is independent,
while (tfitfa)/= tfs/, (£/itfi)/=E 6V-
Finally, if
none of the new transformations are independent of them; for
The case where none of the new transformations are linearly inde
pendent of the old ones is of special interest. If r linearly inde
pendent infinitesimal transformations U\ft U^f, •••, Urf have the
property
(104) (£W)/= ^,-1^7+ ^l/2^/+ - + ^-,P6;/, (/,/= i, 2, ... r),
where the «'s are constants, the aggregate of these and all the trans
formations Uf^L <*\U\f + &tU*f •*? "• -\- aTUTf where these fl's are
any constants constitute an r-parametcr group of infinitesimal trans
formations.*
Remark i. — An r~ parameter group of infinitesimal transformations
is determined by any r of its transformations which are linearly inde
pendent, since the symbols of all its transformations can be expressed
linearly with constant coefficients in terms of any r independent ones.
Moreover it is readily seen that any set of r linearly independent
transformations of the group have the property (104).
* In Note VI of the Appendix an r-parumeter continuous group containing both
finite and infinitesimal transformations is denned. The intimate relation between these
two classes of groups is brought out in Lie's Principal Theorem at the end of the Note.
148 TIIKORY OF DIFFERENTIAL EQUATIONS §43
Turning our attention now to the 'transformations which leave a
differential equation of the second order unaltered, we shall first prove
THEOREM I. — If a differential equation of the second order is in
variant under U\f and U^f, it is invariant under (U\U»)f.
For, if 6^/and U^f leave
(98) ? = F(*,y,f)
unaltered, the extended transformations £7^ and Ujf leave
x ,,
unaltered. By the theorem of § 37, (£//£/:/)/ or its equal
(see Note V of the Appendix) leaves Af = o unaltered. Hence
Theorem I follows.*
In § 42 it was established that the number of linearly independent
infinitesimal transformations that leave any differential equation of
the second order unaltered is limited. If in the case of a given
differential equation this number is r, all the infinitesimal transforma
tions leaving the differential equation unaltered are linear functions,
with constant coefficients, of any set of r linearly independent ones
UJ, #>/, •••, Urf. By Theorem I (£/£/)/, for /,/= i, 2, •••, r,
must also leave the differential equation unaltered. Hence they, too,
are linear functions with constant coefficients of the set U\ft U^f, •••,
Urf. The latter therefore have the property (104), and we have thus
established
THEOREM II. — The aggregate of all the infinitesimal transforma
tions leaving a given differential equation of the second order unaltered
constitute an r-parameter group. Here o <>< 8.f
* This theorem is true for a differential equation of any order, and is proved in the
same way.
f The same theorem is true for a differential equation of the »-th order, where n > 2.
In this case o < r < n + 4.
§43 ORDINARY, OF THE SECOND ORDER 149
It is possible that a smaller number than r, say s, of linearly inde
pendent infinitesimal transformations in an r-parameter group deter
mine a group ; the latter is known as an ^-parameter subgroup of the
larger group.
The four transformations
determine a four-parameter group ; for they are linearly independent, and besides
(UzU*)f=Utf- L\f, (U*Udf=-U*
Of the subgroups of the four-parameter group the following are immediately
obvious :
The two-parameter subgroups i7\f, U»f\ U\f, U,\f; U\f, U\f\ U^f^ U\f\
Uzft U*f.
The three-parameter subgroups L'\/, U»f, U\f\ U\f, L ':{/, L\f.
Uf^x^-—y^L=U\f— U\f, also a transformation of the four-parameter
group, determines with U^f and U$f a three-parameter subgroup, since
= 2 U-,f, ( U.£ 6s)/= - Uf.
Remark 2. — Starting with two or more linearly independent infini
tesimal transformations which leave a given differential equation of
the second (or higher) order unaltered, a group of infinitesimal trans
formations is determined which is either the r-parameter group of
Theorem II or a suogroup of it.
For, let £/i/, £/2/, •••, UJ, (2<k<r) be a set of linearly inde
pendent transformations which leave the differential equation of the
second order unaltered. By Theorem I, (£/|£/)/, (/,/= i, 2, •••, X')
also leave the differential equation unaltered. Some or all of thq^e
may be linearly independent of the original ones. Let /:' of tin in
be such. We know that k + k' < 8. Adding these to the original set,
combine the larger set in pairs by the alternating process as before.
150 THEORY OF DIFFERENTIAL EQUATIONS §43
The resulting transformations also leave the differential equation un
altered. If any of these are independent of the members of the
larger set, add them to the latter, thus forming a still larger set of
linearly independent transformations leaving the differential equation
unaltered. Proceed with this set as before. Obviously this process
must be a finite one, since the maximum number of members of a
set is eight. So that the above process stops when no new trans
formations independent of the previous ones arise as a result of the
alternating process. If the number of independent transformations
finally appearing is r, the r-parameter group determined by them is
precisely that of Theorem II ; if the number is s < r, the ^--parameter
group determined by them is a subgroup of the other.
We shall prove
THEOREM III. — Every r-parametcr group (r > 2) contains tivo-
paramcter subgroups.*
As a matter of fact we shall show that, fixing upon any one of the
transformations, say UJ~, a set of r — i constants c.,, c3) -••, cr can be
found such that
constitute a two-parameter subgroup ; it being understood that the
r-parameter group is determined by 6^/, U,f, ••• Urf, which are,
therefore, subject to the conditions
r t
(104) (Utf)f= ^a,JkUkf, (ij= i, 2, .-., r).
k i
In order that this be the case
(fsV«/+ c,UJ
-f -. +
* This theorem and its proof hold, without modification-, for groups involving n
variables.
§43
ORDINARY, OF THE SECOND ORDER
Since
(U» M\fmc£UJWmc^aj^ and since
U\f, UJ, "•, Urf are linearly independent, (105) can hold only in
case
(106)
= a
= 2, 3, ~;
j=2
Conversely, if c^ <r3, •••, cr can be found to satisfy r equations of the
type (106), where a and b are any constants, and not all of the <:'s
zero, the group Uf= jVj&^/will determine with UJ a two-param-
13
eter subgroup ; for in this case
That such a set of c's can always be found may be seen as follows :
The last r— i equations of (106) are the linear homogeneous equa
tions
(107)
These can be solved provided b satisfies the equation
(108)
= 0.
This equation necessarily contains b, since the coefficient of br~^ is
(— i)1""1. Using any value of /> satisfying it, the c\ are determined
to within a common factor (which is not essential), by solving (107).
The value of a is then determined by the first equation of (106).
Thus Theorem III is not only proved, but a method for finding the
two-parameter subgroup is also given.
152 THEORY OF DIFFERENTIAL EQUATIONS §§43,44
The transformations
w=f, *v=f^ u*f=y&+*y:
dx dy dx dy
determine a three-parameter group, since
Inspection shows that £/]/and U»f determine a two-parameter subgroup. To
find another two-parameter subgroup of which Us/is one of the determining ele
ments, the method of this section may be employed. The constants c\ and c%
must be so determined that
c, £/»)/= a U-,/+ 6(fl Ui/+ c2 6V) >
i.e. - fi 6V- c» Uif= a U*f+ bc^ 6V + be* Uzf.
.•. a = o, bc\ + c» = o, bc» + c\ = o.
In order that the last two equations be consistent, b must satisfy the equation
ft* —1=0,
whence b = . ± I and — = T I. Hence
c-i
* = ,& + ,%, Uf=&+% and U^, &+JC&,Ufs&-&
d* dy dx dy dx dy dx dy
are two two-parameter subgroups of the original group.
44. Classification of Two-parameter Groups. — If a two-parameter
group is determined by £/L/and U*f,
Either both al and a2 are zero or they are not. In the latter case it
is always possible to find a pair of transformations to determine the
two-parameter group for which one of these constants is unity and
the other zero. For, if #, ^= o
§44 ORDINARY, OF THE SECOND ORDER 153
are linearly independent and
Moreover, if for any pair of linearly independent transformations
and U-J of a two-parameter group
this is true for every pair, since
= o
for all choices of constants clt c2, b^ b^. Hence every two-parameter
group can be represented by a pair of transformations U^f and
such that either
= o or ( UtUftfm UJ.
These two possibilities are mutually exclusive ; any grotip can come
under one head only.
A second mode of classification is suggested by the following :
If a two-parameter group is determined by U^f and U<J which are
connected by a relation of the form
where p(x, y) is not a constant, J every pair of distinct transforma
tions of the group are connected by a relation of the form (109) ;
* If «! = o, rt.j ^fc o, so that ( Ul &y/= a^U^f, the groups
satisfy the condition ( V\ K2)/= V±f.
t It is interesting to note that (£/i £/;>)/= o is the necessary and sufficient condition
that each transformation of the group generated by U^ f be commutative with every
transformation of the group generated by £/2/ For an elementary proof of this fact,
see Lie, DiffcrfntuilglciclniHgen, p. 305.
t While in this case f'j/and ('.,/ are distinct transformations, the one-parameter
continuous groups generated by them have the same path-curves.
154 THEORY OF DIFFERENTIAL EQUATIONS §44
for if
Hence all two-parameter groups may be divided into two classes
according as their distinct transformations are connected by a relation
of the form (109) or not.
These two modes of classification are independent of each other.
Hence four classes of two-parameter groups may be distinguished ac
cording as thc\ are representable by a pair of transformations U^f and
U,f=p(x,y}L\ft
P (x, r)
Classify the following two-parameter groups :
£ + (;-x)Z xf+y
dx dy dx d
Ex. 3. x(x+y)+y(X+y), X(X -y) .+;(* - v) .
OUT OV
. 5. - — ,
x. 6. .v + v,
f).v ' ay
.v+;'
§45 ORDINARY, OF THE SECOND ORDER 155
45. Canonical Forms of Two-parameter Groups. — By a proper
choice of variables the various classes of two-parameter groups can
be reduced to certain simple forms which Lie called their canonical
forms. These will now be determined in turn.
By the method of § 9 a set of variables can be determined so that
Uif takes the form
dx
If the resulting form of U^f is
^EEO and ^ = o, since (U,U*)f= ^ Bf- + ^ ¥ = 0. Hence
ox ox dx ox ox oy
£ and 77 are, at most, functions of y ; /'.<?.
where rj(y) =£ o, since U,f^ p(x, y) UJ.
The transformation ^/,/= -^ remains unaltered by a change of
variables of the type
where <f>(y) and ^(j) are at our disposal. This change of variables
causes £/>/to take the form [(15) § 9]
assumes the form .
If <#> = f^ dy and ^( v) = f ^' , UJ
J iy(>') »/ T;(J)
156 THEORY OF DIFFERENTIAL EQUATIONS §45
Hence, by a proper choice of variables, a pair of transformations
satisfying conditions a can be made to assume the canonical forms
ox dy
Having established the existence of the canonical forms in this
case, the actual finding of the canonical variables (which reduce the
transformations to these forms) can be accomplished by two quadra
tures. For, starting with a pair of groups
+ and
ox oy ox oy
satisfying conditions a, the new variables x and y will reduce these
to the forms ^ /• •) /•
UJ=%- and IV*?
dx dy
respectively, provided x satisfies
f. dx , dx f. dx , dx
and y satisfies fi |» + „ ^ = o, (, f- + „ ?» = ,.
dx dy dx dy
Since such new variables must exist, the equations of each pair
must be consistent. Because Utf^pU\ft they can be solved for
— t JL and — , — respectively, whence x and y are determined
dx dy dx dy
by the two quadratures
x ay
The transformations
§45 ORDINARY, OF THE SECOND ORDER 157
form a pair satisfying conditions (t. From
- + ,•:=, and +*=o
whence x
Similarly, from _>! & + * |* = o and * ft +, f*~ I,
d* 07 d-r ' dj'
5£=— ^— and ^^
^
whence y - = } V
J «• + y2
These canonical variables are obvious from geometrical considerations.
As before, £/J/"can be reduced to the form
by the choice of canonical variables (§ 10). Then t/,/ assumes the
form .,
where O-(AT, y) is what p becomes when the old variables are replaced
by the new. Since
dy dy
cr is a function of x only. Taking this as the new A*, which change
of variables leaves U\f unaltered, £7>/assumes the form .v - • Hence,
ov
by a proper choice of rtiriiifi/t's (7 pair c/ transformations satisfying
158 THEORY OF DIFFERENTIAL EQUATIONS §45
conditions fi can be made to assume the canonical forms
TTf-ty .T,_ d/
v\J = T- , c o/ = A - - •
3;' dy
The actual finding of the canonical variables in this case requires
a single quadrature. For, starting with a pair of transformations
£V=*£ + ,£ and ^/Wf^ + JA
ax dy \~ ox vyj
where ( L\ U^f = UlP UJ = o, i.e. UiP = o,
the new variables x and y will reduce these to the forms
respectively, if x = p (.Y, v)
and y satisfies the equation (§ 10),
4+^=i.
fl.v 3>»
Moreover, since U}p = o, p is a solution of
Hence, y = F(x, }') is some solution of
dx _ dy _ dy
7~7~T
distinct from p = const., which is also a solution of this system of
equations. Among the various ways that will suggest themselves
when £ and r) are given in any specific case, a possible method is to
solve p(.\-, jr) = c for one of the variables, say x = <}>()>, c), whence
y
= f <*y
J r,(<t>,)>)
§45 ORDINARY, OF THE SECOND ORDER 159
The transformations
Uif=xy &. + 2y* V- and U->f=x* ^L + 2*2y ^L
fa dy dz dy
form a pair satisfying conditions p.
y must satisfy xy^> + 2y---y- = i.
d* dy
To solve the corresponding system of ordinary equations
<Xr dy- _ dy
xy 2y* I
use may be made of the solution — = c or y = — • Then
y c
2x 2y
y. (UMV= (/i/, U.J=£p(xt
As before, 6^/can be put in the form
by introducing canonical variables. Taking 6^/in the form
,
we must have ^r — ° an<l ^ = J> since
Hence, i^/must liave the form
160 THEORY OF DIFFERENTIAL EQUATIONS
where £(x) ^ o, because U.^f^pU^f. The introduction of the new
variables
x=t(x)tV = il,(x)+y
leaves U^f= — unchanged in form, but changes cZ/into
d
dx dy
This takes the form Uo/= x-~ +y^-
dx By
tdjc
when £(x)$(x) = <j>(x), or <j>(x) = e]*W
r_
and £(*}$' (x) +X(x) =t(x), or if/(x) = -e^Mj ^ e~^(^ a'x.
Hence, by a proper choice of variables a pair of transformations
satisfying conditions y can be made to assume the canonical forms
The actual finding of the canonical variables in this case requires
two quadratures. For, starting with a pair of groups
satisfying the conditions y, the new variables x and y will reduce
these to the forms
respectively, provided x satisfies
t dx . dx t dx . dx
£15- + *?!^ =°> ^T- + ^--
ox dy dx dy
§45 ORDINARY, OF THE SECOND ORDER l6l
and y satisfies
£ dy dy_ fdy dy _
ClT--r »/lT-= I? C2-T- -I- >?2T- — U-
dx dy dx dy
Since such new variables must exist, the equations of each pair must
be consistent. Because U-2/^pU\f they can be solved for — •, —
-» a dx dy
, dy dy
and -^, -^, giving
dx dy
__ ft
djt ^772 — ^r/! ^172 — ^i ^J ^1^2 — £2171 Utyi — tL-^i
3 log x j d log JT
Dividing (no) by JT, - «•- and - « — are given, whence log JT is
ox dy
obtained by a quadrature and the form for x follows.
Equations (in) maybe solved in various ways. The most gen
eral form for y satisfying them is not needed. As a matter of fact,
the simpler the form obtainable, the better. One way of proceeding*
is to assume that x and y are no longer independent, but that y = ex
where c is a constant. Then
(112) ^=^+^ = (X + ^)i/ + ,+ ^
ax ox oy
where X, /x, v, IT are what the corresponding coefficients in (in)
become when y is replaced by ex. Since (112) is a linear ordinary
differential equation of the first order, it may be solved by the usual
method, involving two quadratures (EJ. Dif. /£/. $j 13). A process,
however, by which a single quadrature alone is involved in solving
(112) is given by the following :
* Special methods will frequently be found simpler, however.
1 62 THEORY OF DIFFERENTIAL EQUATIONS §45
Inspection of equations (no) and (in) shows that y=x(x, ex),
which is obtained from x(x, y) by replacing y by cx9 satisfies the
equation
Hence the transformation y = vx(x, ex) reduces (112) to
dv _ v -f- CTT
dx x(x, ex)'
whence v is obtained by a quadrature. Then y follows at once, after
replacing c by -L.
x
The transformations
and U»f=x'i + (y + xy)
dy dx dy
form a pair satisfying the conditions 7. From
=0 and x*-\-
dy dx dy
.
**' dy
= i
From x&=i and ^ ^ + (y + xy) m y,
dy dx dy
fa - y. -.y^rj^L ^-i.*
dx x* *J ' dy x
Putting^ = cx -7~^i~~i
dx x1 x1
* These equations can be solved directly. From the second one y=-
where 0(.r) is to be determined. Putting this value of y in the first equation gives
_\
4&_£, whence* •* *.
dx x*
k = Q gives the form for y obtained in the text.
§45 ORDINARY, OK TIIK SWOND OKDI.R 163
1
Using the method given above, put y — ve x. The linear equation then reduces
i i
to — = — — ex ; whence v — cex.
(/JC X
8.
As before, by the introduction of canonical variables £/,/ can be
made to assume the form
These variables will cause £72/to assume the form
Since ( £/, «,)/ = UJ, = i and r, = A'(.v) +y. So that
The change of variables y = X(x) + v leaves ^/unaltered and
changes U*f to the form U.,/ = y~-' Hence, f>y a proper choice of
variables a pair of transformations satisfying conditions 8 can /><-
made to assume tJie canonical forms
The actual finding of the canonical variables in this case requires
the solving of the differential equation of the first order determining
the path-curves of the group generated by either of the transforma-
164 THEORY OF DIFFERENTIAL EQUATIONS §45
tions. For, starting with the pair of transformations,
where (U^U^f = L\f, the new variables x and y will reduce these to
the forms .,, . ,
respectively, if y — /> ( JP, j) *
and x satisfies the equation
.dx dx
^+7^=°-
The solution of this equation is usually obtained by first solving
dx dy
T"T'
the differential equation of the path-curves of £/,/.
The transformations
,T/^_4/anW-|_,|'
form a pair satisfying conditions 5.
y
The solution of ^ + f£/ = oisi + - = const.
Ex. Determine the canonical variables for the groups at the end
of § 44.
* The other requirement of y, viz. Uy= i, follows from the given conditions on U^f
and i/a/ since (U^
§46 ORDINARY, OF THE SECOND ORDER 165
46. Differential Equation of the Second Order Invariant under Two
Groups. — Starting with two non-trivial infinitesimal transformations *
which leave a differential equation of the second order
unaltered, an .r- parameter group of infinitesimal transformations,
leaving (98) unaltered, can be found (Remark 2, § 43), which con
tains a two-parameter subgroup (Theorem III, § 43) determined by a
pair of transformations ^/and U2f which satisfy one and only one
of the conditions (§ 44),
(l/M)/=o and (UM)/= UJ.
Moreover, these two transformations can be found by direct and
practicable processes from the original two transformations, and they
also leave the differential equation (98) unaltered.
We shall now suppose that we have found such a pair of infini
tesimal transformations U\f and U^f. Passing, as was done in § 39,
to the corresponding linear partial differential equation
the latter is invariant under the extended transformations U\J and
UJf, which are subject to one of the conditions
(£//£/,')/ EEO and (^'6/a')/
since (f//c7/)/=(^c72)7 (see Note V of the Appendix). Two im
portant cases are to be distinguished :
A. A relation of the form
(970 UJf^aWf + pAf
* As use is to be made of the properties of groups of infinitesimal transformations,
the one-parameter groups under which (98) is invariant will be replaced by their repre
sentative infinitesimal transformations in what follows. (Compare Remark, § 6.)
1 66
THEORY OF DIFFERENTIAL EQUATIONS
§46
exists. In this case UJ and cZ/ determine distinct path-curves,
that is, no relation of the form
(109)
can connect them. For, if such a relation did exist, and if
Uzf would have the form
f do- . do- f > d<r , .do- ,,\3/
+*-;£+**;,>' -feV^'-^ £•
A relation of the form (97') implies the vanishing of the determinant
— ,
dx \' By
do-
dy
This reduces at once to
c)cr ,. do-\ ,
^ r*~ ^ T~ /
^ 5^/
,. d(T ,.,"1
•*• « ^-/
qy
Since neither 171 — ^7' nor both ^ and ^ can be zero identically,
A can vanish identically only in case ~ = o and -— = o simultane-
O.T dy
ously, that is, <r must be a constant. This would make U\f and
one and the same transformation. Hence the relation (109) cannot
hold when (97') does. Assuming that (97') holds, two cases must still
be considered :
§46
ORDINARY, OF THE SECOND ORDER
I67
By means of two quadratures (§ 45, «) canonical variables can be
found so as to reduce the two transformations to the forms
dx
-
By
respectively. Since the differential equation expressed in terms of
these variables must be left unaltered by these two transformations, it
must be free of both x and y (I and I', § 28). Hence it has the form
and the corresponding partial differential equation has the form
loreover,. _„ _ ^ _ _^
The relation (97') implies that
i y
I O
O I
= F(y') = o.
Hence when conditions (97') and i° hold, the introduction of canoni
cal variables for the two-parameter group reduces the differential
y" = o,
y = ax -f /;.
equation to tJie fonn
and the solution is
By means of two quadratures (§ 45, y) canonical variables can be
found in this case, reducing the transformations to the forms
dx
1 68
THEORY OF DIFFERENTIAL EQUATIONS
§46
The introduction of these variables reduces the differential equa
tion to the form (I and IV, § 28)
', xy")=o, or y"=
The corresponding linear partial differential equation has the form
Moreover,
The relation (97') implies that
, y'
O I
x y
EE - F(y') = o.
Hence, also, in the case where conditions (97') and 2° hold, the
introduction of canonical variables for the two-parameter group
reduces the differential equation to the form.
y" = o
and the solution is y = ax -+- b.
B. No relation of the type (97') exists. That is,
Here the two subcases in A are also to be considered.
Since this carries with it
§§ 46, 47
ORDINARY, OF THE SECOND ORDER
I69
the conditions of § 38, B, 4°,(#) exist, and the two solutions of the
corresponding linear partial differential equation are given by the
two quadratures
dx dy
i y
£1 >?i
where
AEE
= # and
/
I
y
Eliminating 7' from these gives the solution of the original differen
tial equation.
Since this carries with it
the conditions of § 38, B, 4°, (<£) exist. Two solutions of the cor
responding linear partial differential equation are obtained by two
quadratures, by the method given there. Eliminating y' from these,
the solution of the original differential equation follows.
Remark. -•- It may be noted that in every instance where an
ordinary differential equation of the second order is known to be
invariant under two distinct groups, of which neither is trivial, its
integration can be effected by means of two quadratures.
47. Second Method of Solution for B. — The method in cases A,
i° and 2° of the previous section leaves nothing to be desired. For
the remaining cases, however, while, theoretically, the reduction of
the problem to two quadratures seems sufficiently simple, a method
analogous to that employed for A, even if involving a larger number
1 70 THEORY OF DIFFERENTIAL EQUATIONS §47
of quadratures, or possibly the solution of a differential equation of
the first order, may prove simpler in actual practice. Still under the
supposition B, viz.
the four possible forms (§ 44) of the two-parameter groups of infini
tesimal transformations leaving the differential equation unaltered
will be considered :
By a process involving two quadratures (§ 45, a) canonical varia
bles x and y can be found, reducing the infinitesimal transformations
to the forms
The differential equation invariant under these has the form
(I and I', § 28)
An additional quadrature gives
/rfy'
F(y')
or, when solved for y', y' = <j>(x -f #),
and a final quadrature gives the solution
y = I (f) ( x -f- (7 ) tix -f /^.
In this case four quadratures are required.
/,)/EEo, U,/=P(.r,y)tyj.
§47 ORDINARY, OF TI1IO SKCOND ORDFR I/I
By a process involving one quadrature (§ 45, /j) canonical varia
bles x and y can be found, reducing the infinitesimal transformations
to the forms
The differential equation invariant under these has the form
and VII, J ,8) yll = F(^
Two additional quadratures give the solution
y =f CF(X) dx- + ax + b.
In this case three quadratures are required.
By a process involving two quadratures (§ 45, y) canonical varia
bles x and y can be found, reducing the infinitesimal transformations
to the forms
The differential equation invariant under these has the form (I and
IV, § 28)
As in the case a, two additional quadratures give the solution.
In this case four quadratures are required.
8. (UMV^Ui/, 6i/=PCv,v)^i/
By a process (§ 45, 8) involving the finding of the path-curves de
termined by either infinitesimal transformation, i.e. the solution of
the differential equation .
1/2 THEORY OF DIFFERENTIAL EQUATIONS §47
canonical variables x and y can be found, reducing the transforma
tions to the forms
The differential equation invariant under these has the form (I and
HI, § 28)
Two quadratures give the solution
Remark. — The above classification holds equally well for A, for
which it is exceedingly simple, cases ft and 8 never arising (§ 46).
Hence the method of introducing canonical variables applies to all
cases where a differential equation of the second order is invariant
under two groups. The interest in § 46 lies in the fact that it is there
shown that it is always possible, if desirable, to solve the differential
equation by two quadratures only.
While the classification of § 40 is more complicated, it must be
borne in mind that the two groups employed there need not deter
mine a two-parameter group. Some of the methods of § 40 are ex
ceedingly simple ; so that they are not to be ignored. On the other
hand, it is suggested that the method of this section be applied to the
examples of § 40.
Ex. 1. xyy" + xyn-yy' = o. (Ex. i, § 40).
This equation is invariant under U\f= x-J- -f- y •£• and £Z/= v -•
dx oy ' dv
These determine a two-parameter group of the type «. The canoni
cal variables are readily found to be x = log*, y = log^. Introduc-
X
ing these, the differential equation takes the form
§47 ORDINARY, OF THE SECOND ORDER 1 73
Integrating this, one obtains
log -~^ + '* = '°r if = T^F— : = ~^»'
Integrating again,
2 y + c = log (^ - e >2X) or ^ = ^ - «T2jr.
Passing back to the original variables^
^2-^/ = i.
Ex. 2. /' = /y + Qy + X. (Ex. 2, § 40) .
This equation is invariant under £7i/= V\— and K/syf-^, if
5y " dy
j/' = /'v/ + <27i and >-,/' = Pyi -f fe- The transformations UJ
and ^/determine a two-parameter group of type /3. The canonical
variables are x = ^, y = 2-. To introduce these use should be made
*
of the fact that = = - where A = \\ y./ - y2ylfj and that
^r </or dx y\2
y\y* '- -y*yi ' = /^. Then
Substituting these values in the differential equation gives
where the right-hand member must be expressed as a function of x.
Integrating twice,
174 THEORY OF DIFFERENTIAL EQUATIONS §47
Passing back to the original variables,
t>) f
iA'
,
(AX* -jyi )
. — It is an interesting fact that this form of the solution
includes as a special case the form obtained by a well-known method
in case the coefficients in the linear equation are constants. (See
EL Dif. Eq. § 47-)
Ex. 3. /' = // + Qy. (Ex. 3, § 40.)
This equation is invariant under U^f=)\-- if y1 is a particular
solution of the equation, and also under R/Byu. The trans-
dy
formations C^/and U*f determine a two-parameter group of type 8.
The canonical variables are x = x, y = 2-. This change of variables
y\
is the one usually employed. (See EL Dif. Eq. § 53, i°.)
CHAPTER VII
CONTACT TRANSFORMATIONS
48. Union of Elements. — The configuration consisting of a
point and a line * through it is known as a lineal clement. It is
obviously self-dualistic. Since a lineal element in the plane is deter
mined by three coordinates,! there are oo ;>> such elements.
Any curve in the plane determines co J lineal elements, each one
consisting of a point of the curve and the tangent line at that point.
[In particular a straight line determines oo J lineal elements, all hav
ing the same /-coordinate ; while a single point (looked upon as a
line curve of the first class) determines co 1 elements all having the
same x- and jv-coordinates]. Such a single infinity of lineal elements
is said to form a union of clements,\ and successive elements in this
case are said to be united. In general co * lineal elements do not
form a union ; it is easy, however, to find the condition that they do :
Two relations among the three coordinates
(113) 0(,r, y, /) = o and i//(.v, y, /) = o
* At times it is convenient to replace the line by its direction in the above definition.
t We shall use the nonself-dualistic set (x, y, /) where x and y are the rectangular
coordinates of the point and / is the slope of the line.
It is almost needless to add that the theory here developed is no more restricted to
this choice of coordinates than the general theory of Analytic Geometry is confined to
the use of Cartesian coordinates.
J In this case the locus of the points of the elements coincides with the envelope
of the lines of the elements ; and besides, the point of tangency of each line with the
envelope is the point of the element to which the line belongs. This locus will be re
ferred to as the curve of the union.
175
1/6 THEORY OF DIFFERENTIAL EQUATIONS §48
determine oo 1 elements.* The locus of the points of the latter
(114) o>O,/)=o
is obtained by eliminating/ between the two relations (113). A
union exists provided the value of/, in terms of x and y, obtained
from either of the two relations is the same as that of the slope of
the tangent to the curve (114), i.e.
*"*~~~5'
where partial differentiation is indicated by a suffix. The condition
that the lineal elements determined by (113) form a union is therefore
that
(JI5) dy
Ex. 1. Starting with the relations
the point locus is the circle x1 + jy2 = I. Here
£=-*=/.
dx y
FIG. 6 Hence the elements form a union. (See Fig. 6.)
* A single relation (j>(x, y)=o free of/ defines ool unions, each consisting of
the oo l elements having a point of the curve 0 (x, y) = o in common, p being un
determined.
Hence, if neither of the relations <f>(x, y) =o and \f/(x,y) = o involves p, they to
gether determine a finite number of unions, each consisting of the oo » elements hav
ing in common a point of intersection of the curves (f>(x,y) = o and \f/(x,y) = o. (See
Ex. 4, below.)
f The same condition obviously holds when the lineal elements are determined
parametically
("6) * = *(/), y= Y(f),p = p(t).
CONTACT TRANSFORMATIONS
177
Ex. 2. In the case of
y + xp = o, x + yp =o
the point locus is the pair of lines x2 — y1 = o.
Here -2- = - ; while / = — - • Hence there is no
dx y y
union. (See Fig. 7.)
Ex. 3. In the case of
y = xp -\- i, p — a = const.
the point locus is the line y — ax -|-
— a — p. Hence the elements form a union. (See Fig. 8.)
FIG. 8
Here
dx
Ex. 4. In the case of
the point locus is the point x = o, y = i, while / is undetermined. The
elements form a union. (See Fig. 9.) FIG. 9
Ex. 5. The elements determined by
x = cos /, y = sin /, / = tan /
do not form a union, since the point locus is the
circle x'2+y*=i, where
dx y
(See Fig. 10.)
FIG. 10
Ex. 6. In the case of
y-
the point locus is the line y — I. Along this p = O.
Hence the elements form a union. (See Fig. n.)
FIG. ii
THEORY OF DIFFERENTIAL EQUATIONS §§48,49
Ex. 7. In the case of
y = xp+l, 7 = 3
the point locus is the line y = 3. Along this p = o.
But the elements along this line determined by the
first relation have p = *- — — ^ o. Hence there is
I2 no union. (See Fig. 12.)
49. Contact Transformation. — Of the possible transformations
on the coordinates of a lineal element
(117) Xl = X(x, y, /), y, = Y(x, y, /), pl = P(x, 7, /) ,
those which transform every union of elements into a union play an
important role and are known as contact transformations. • The
condition that (117) be a contact transformation is readily seen
to be
( 1 1 8) dy\ - /! dxi = p (x, y, p) (dy —p dx) , where p =£ o.
For, from the condition (115) it follows that if a union is to be
transformed into a union dy\ — p\ dxv must vanish whenever dy — p dx
does ; that is, the former must contain the latter as a factor.
Indicating partial differentiation by a subscript, (118) may be
written
This is equivalent to
(119) Yf-PXt=-pp, y,-JPX,
whence
(,20) J'=
§49 CONTACT TRANSFORMATIONS 179
and
(121) Xlt(Y,+pY,)- Y,(Xf+pX,) = o*
The two relations (120) and (121) may be put in the compact
V V i j, V
Xp X* +P Xy
These relations, which are necessary conditions that (117) be a
contact transformation, are also sufficient, as may be seen as follows :
They lead at once to
Y — PX Y — PX
or *•*— ' A* = £L — £21i = p.-j-
-/ i
Equations (119) follow at once, and, therefore, condition (118)
is fulfilled.
Conditions (120) and (121), or their equivalents (122), may thus
be used instead of (118), when desired.}:
* Introducing the Poissonian symbol
r,n_ ** x,+px,\
= yr y.+tr,\-
the relation (121) takes the simple form
(121)
When two functions .Y and Y satisfy the condition (121), they are said to be in
involution.
f This value of the common ratio p cannot be identically zero, for using (122) it
may be written
p = XpYy — XyYp = XxYy — XyYX = Xx Y,, - A'r Y.r .
XP \r pX, ' pXp
all three of thr numerators cannot vanish simultaneously since .Y and Y are supposed
to be independent rum-lions.
J An dement transformation, which is not a contort transformation, transforms pre
cisely oo- unions into unions. (See K;IMI<M, American Journal of idathematics,
Vol. XXXII, p. 393). Thus, A" = .v, Y = />,/' v, which is obviously not a contact
transformation, transforms the union defined by y + p — c^e*, y—p — c.^-1 for any
pair of values of c± and c.± into a union.
180 THEORY OF DIFFERENTIAL EQUATIONS §49
Remark i. — Of the three functions X, Y, P in the contact trans-
«
formation (i 1 7), either one of X and Y may be selected at pleasure ;
the other one is then determined as a solution of the linear partial
differential equation (121). With X and Fselected, P is determined
uniquely by (122).
The extended point transformation (§ 13) is evidently a special
case of a contact transformation. For if X and Fare any functions
free of/, (121) holds; while the form for the accompanying trans
formation ofjv' or/, given by (21), is exactly (122).
In what follows we shall exclude extended point transformations
from consideration, unless specific mention is made to the contrary.
As an example of a contact transformation may be mentioned the transforma
tion by reciprocal polars with respect to a conic. The transformation, in case the
conic is the circle x~ + y1 = i, takes the form
Here _
y(y - xp}
The transformation by reciprocal polars with respect to the parabola x2 = zy
is given by
(-#) *\ —p-, y\ = xp — /, p\ = x.
Here dyi—p\dx\=—(dy — pdx}.
In the above illustrations a union whose curve is a point is trans
formed into one whose curve is a straight line. That in the case of
every contact transformation (not an extended point transformation)
a union whose curve is a point \ must be transformed into one whose
* These equations may be obtained as follows : The point (x, y} of an element
(x, y, p) is transformed into the polar line xx± -\-yy\ = i whose slope is p± = • The
line of the element, ?
Y-y=p(X-x) or ^2*-\ £— = i,
y-xp y-xp
is transformed into the pole
~ y — xp' yi y — xp
\ Excepting possible special points ; e.g. the origin in Ex. 3, p. 185.
§49 CONTACT TRANSFORMATIONS l8l
curve is an actual curve may be seen by eliminating / and p\ from
equations (117). There results from this elimination a single relation,*
(123) F(x,y,x\,yl) = °,
which determines a locus for the points (#„ y^ corresponding to a
fixed point (x, y)."\
Moreover, a contact transformation is determined by a relation of
the type (123), provided the three equations
(124) F=o,
can be solved for x, y, p, and also for xlf y\, JV For, solving for
#i» y\i P\» there results the transformation of the three variables
(117) x, = X(x, yt p), y\ = Y(x, y, /), p, = - ^.
F*
That this is a contact transformation may be seen readily. For
from F F
*—!?•*> — %
* If there were two independent relations,
Fl(*,y, *i.^i)= o, F*(xty, JT,,^) = o,
they could be solved for x\ and y\ in terms of x and^y, which would imply that (117)
was an extended point transformation.
f We may say (fixing our attention on the curve of a union) that the effect of the
contact transformation is to transform any point (a, l>) into the curve F(a, b, x^,y^) = o;
while a point transformation transforms a point into a point.
Moreover, it is not difficult to show that a contact transformation, in general, trans
forms a union determined by a curve C into one whose curve O is the envelope of the
curves into which it transforms the various points of C, or, using the same form of ex
pression as above, we shall say that it transforms the curve C into O. (Thus sec Lie,
Berllhrungstransformationen, p. 49) . If it should happen that the curve C is one of the
curves F (x, y, (t, ft) = o, where Ot and /3 are any constants, its transform C' is the
point («, /3).
1 82
TIIKOKY OF DIFFERENTIAL EQUATIONS
§49
Differentiating (123) gives
FXi dx± + FVi d)\ = — (Fx t/x -f- Fy ffy) .
p
Hence dy\— p\dx± = — -JL(ay—'paxyt
f*i
which proves that (117) is a contact transformation.
The condition that (124) be solvable for x, y, /and for A-,, y^ ft
can be expressed very simply analytically :
In order to be able to solve for/ it is necessary arid sufficient that
Fy-=£o when F— o. Similarly, F =£ o when F = o is the condition
that one be able to s%lve for ft.
The condition that the first two equations of (124) can be solved
for ^ and }\ is the non-vanishing of the functional determinant
or
In the latter the factor - ----- is omitted since it is not zero whenever
F= o, because F is supposed to be generally analytic, and besides
it is not infinite since Fv=£o when F= o, by hypothesis. This de
terminant can be put in the more symmetrical form
A =
Fx Fy
F F F
!/i xy. * y
Since A contains Fas a. factor whenever either Fy or Fv does, the
non-vanishing of A when F — o assures the non-vanishing of Fu and
F . Hence the only condition that (124} be solvable for x^ ylt ft is
(125)
o when F=o.
Because of the symmetry of A as to x, y and x\,y\, (125) is also
the condition that (124) be solvable for x,y,p.
§49
CONTACT TRANSFORMATIONS
183
Remark 2. — It is interesting to note that A =£ o is the condition
that F(x, }', .TJ. _)',) involve x and y as two essential parameters, and
also that it involve xl and )\ in the same way ; when such is the
case F(a, b, xlt }\) = o defines oo2 curves for all choices of a and ^,
and F(xty, a, ft) = o defines oo2 curves for all choices of a and /?.
For if x and y are not essential parameters in F(xt y, x\, y^, two
functions of x and y, say xi(x> }') an<^ Xz(x>y)> can be found such
that (see Note VII of the Appendix)
This carries with it
.•. A =
o F, Fv
\ FXVi FUVi
Conversely, if A = o
f
^ ~\~ \->F F
o.
where p is a constant as far as xl and j^ are concerned, but may be a
function of x and y.
Hence A = o carries with it a relation of the type
Fz — p (x, y)Fu = o,
which is the condition that .v and y are not essential parameters in
F(x, y, *„;'!).
In exactly the same way it can be shown that if A = o xl and Vi are
not essential parameters in F.
1 84
THEORY OF DIFFERENTIAL EQUATIONS
i.,49
The equations of transformation (<-/) and (B) in the cases of transformation
by reciprocal polars given above are readily obtained by the method here given
when jcx\ + yy\ = 1 and xx\_ — y— ji = o, respectively, are selected as the
relation (123).
For the transformation by reciprocal polars with respect to the general conic
ax* + 2 hxy + by'1 + 2 gx + 2fy + c — o
the relation (123) is the equation of the straight line
fi(xyi + yxfi + %'i + g(x + *i) +/(}' + yd + c = O,
(ax + hy + g)xi +(hx + by + f)y\ + gx + fy -\-c-O.
Here
AfEE
o ax\ + Ay i
ax -\- hy -f- g &
hx + by +f h
by\. + f
Subtracting ^ri-times the second row -f >'i-times the third row from the first,
and taking account of (123')
gx +fy + c g f
a x + hy + g a h
hx + by +/ h b
< S f
g a h
f h b
a h g
h b f
S f '
i.e. A equals the discriminant of the conic, and is different from zero in case the
conic is an actual one and not a pair of straight lines. In this case the method
given above applies. Solving
(ax + hy
(a
+(hx + by+f}yv + gx + fy + c = o,
+ (h + bp}yi + g + fp = Q,
ax + hy +g+(hx + by +/)/i = o,
for x\,
i, the formulae of transformation are
_G(xp-y)+ff-Ap _/-(xp-
~ ~
_
p> * kx+ty+f
where A, B, C, F, (,\ //are the respective cofactorsin the discriminant of the conic.
The transformations (.4) and (/?) are obviously special cases of (C).
§§49,5° CONTACT TRANSFORMATIONS 185
Another interesting contact transformation is obtained by selecting for (123)
the equation of the circle,
(123") O - xtf + (y-ytf = r* ^ o.
In this case A =— 8 r2, and the equations of transformation are readily found
to be
(Z?) *i = x ± — & — . yi =y T
Vi-f/- vi +/1
The effect of (D) is to transform any curve into a pair of parallel curves, one
on each side of the original one, and at a distance r from it, as is apparent from
the nature of (123"). A transformation of this type is referred to as a dilatation.
Find the contact transformations determined by the following
relations :
Ex. 1. (x-xiy-2a(v-yl)=o. E^ 4 ^i+^i=I>
Ex. 2. ^^iT + IZ^i)!=I.
Ex. 5. *+*«!.
•*! Jl
Ex. 3. av' + Vf-Ov^ + ri'O^o.
50. Group of Contact Transformations. Infinitesimal Contact
Transformation. — If in the one-parameter group
(126) xl = X(xtytpt a), y,= Y(x,y,p, a), p, = P (.v, .v, /, a)
the condition
(i 1 8) <iY-PtiX = P(x, y, p) (//r -/./jf),
or its equivalent (122), holds, (126) defines a ant-parameter group of
c 'en tact transformations*
Like any one-parameter group in three variabK-s ($11) the group
(126) contains an infinitesimal transformation
(127) .V! = A' + ^(A-,.V,/) oa, v, = v + iy (.v,.v,/) Sr/,/, =/-f-7r(.v,
1 86 THEORY OF DIFFERENTIAL EQUATIONS §50
whose symbol may be written
<'*> ^€+'f^f-
Thus the dilatations
form a group, with the infinitesimal transformation
£y- p df _ I
^T+J1 d* Vi
Similarly the transformations (Ex. I, § 49)
*i = *— «/, ;'!=/- ^ ,
form a group, with the infinitesimal transformation
Since (127) is also a contact transformation,
(i 18') d\\—pi dx^—dy—p dx+ (dy — p <% — TT <tx)$a*s=£p(dy—p<&).
.-. p = i + a- (x, y, /) &z,
where
( I 29) <r(x, ;-,/) (ity—p<tx) =<h)—p<%—irt/x = ff(ii—p£) -f^ t/p — irt/\.
Writing with Lie "
(130) r)-/>£= - /r(-v» .'',/),
where /Fis known as the characteristic function of the infinitesimal
contact transformation, the identity (129) may be replaced by
//', + TT = o/, - ^Fv = IT, - /Fp + ^ = o ;
whence, making use of ( 1^0) and eliminating <r,
* Here, as alw.ivs it) tin- case of infinitesimal transformations, higher [.ou.-rs of Sa
are nrtjlcctcrl.
§5o CONTACT TRANSFORMATIONS 187
Moreover, for all choices of the function IV, £, rj, TT are so de
termined by (131) that the corresponding infinitesimal transforma-
satisfies the condition (118') and is therefore a contact transforma
tion ; hence the
THEOREM. — Connected with every infinitesimal contact transforma
tion t)ii' re is a cJiaracteristic function IV = — 77 + /£, in terms of
which the transformation is given by means of (131). Conversely,
starting with any function IV, the relations (131) define an infinitesi
mal contact transformation.
In terms of the characteristic function the infinitesimal transforma
tion takes the form
(132) /i/= \vt ^+ (Pwr- w)V-( w,
or using the Poissonian symbol (§ 49)
(13*) Bf
Choosing for IV the form
gives the infinitesimal transformation
Bj- P df
Vi +/5 ~d* v
which belongs to the group of dilatations (/?).
The selection
gives the infinitesimal transformation
**f— 7- -\
which belongs to the group
*"=.,»=,- ..,,/, = /. CK*.».H9)
/>-
1 88 THEORY OF DIFFERENTIAL EQUATIONS §50
When W is linear in /, the corresponding transformation is an
extended point transformation. For, if
[(24), § 13]
Another fact worthy of mention in connection with the character
istic function is the effect upon it of a change of variables when the
latter is effected by means of a contact transformation. As was
noted in § n, the introduction of the new variables
[14'] x = F(xty,p), V = *(xty,p), P = *(x,y,p)
causes the infinitesimal transformation (128) to take the form
By the definition o_f the charactertisic function (130) its form
after transformation is
- yp).
If [14'] is a contact transformation,
dy — pdx=. p(dy — p dx) ,
or yx-pxx=-Pp, yu-pxy = p, yp-pxp = o;
whence
(133) W(x, y, p) = p(tf-ri) = P(.v, v, /) lV(x, .v, p).
Of course, in the right-hand member, x. v, p must be replaced by
their values in terms of the new variables given by [14'].
§§50,51 CONTACT TRANSFORMATIONS 189
The characteristic function for the group of dilatations in the case of rectangu
lar coordinates was seen to be
w
Introducing the new variables, '
for which dy — p dx = — (dy — p dx~),
it is easy to verify that
W = - VTT/ = - Vi + x*.
On the other hand the new variables
y - xp y — xp y
for which dy—pdx = — (ify — p dx)
y(xp - .i')
cause the characteristic function to assume the form
W = -fi±£ = (xp - y) Vor* + j
51. Ordinary Differential Equations. — A differential equation of
the first order
(i34) f(x,y,p) =o
may be looked upon as a relation among the three coordinates of the
lineal elements of the plane, with the understanding, however, that
(•35) P = di-
dx
So that the differential equation defines oo2 lineal elements which
[because of (135), which is identical with (i 15), >$ 48] are arranged
in oo1 unions. The solutions of the differential equation are the
equations of the curves of the unions.
Since all the lines through a point constitute a union, in which
case the common point is the curve of the union, such unions must
be taken into account when looking for the solutions of a differential
IQO THEORY OF DIFFERENTIAL EQUATIONS §51
equation. Thus if the relation (134) is free of/, say
this may still be looked upon as a differential equation in which / is
arbitrary. Such a differential equation defines, besides the union
whose curve is /"(.*, y) = o, those unions
determined by each of the various points
• ^V--\ of the curve. See Fig. 13. Each of
these points will be considered as an
p integral curve of the differential equa
tion.
Since every lineal element of the envelope of a family of <x> l curves
is an element of some curve of the family (compare EL Dif. Kq.
$$ 29, 30), the equation of the envelope must also be a solution
(i.e. the singular solution) of the differential equation of the family of
curves. In the special case of a differential equation of the type
(134') the curve f(x, y) = o maybe looked upon as the envelope,
and its equation is therefore the singular solution.
The Clairaut equation (El. Dif. Rq. § 27)
when transformed by
(/?) jf! —p, y\ = xp — y, f>i = x,
takes the form y\ +/(•* "0 = °»
which is of the type (134'). It has for integral curves the various points of the
curve vi -f /(•*"!) = o, while the equation of this curve itself is a singular solution.
Passing hack to the original variahles, this curve is transformed into some curve
<()(jc,y} = o, and its points are transformed into the tangents of 0f.r, y)=O.
Their respective equations are the singular and particular (in the aggregate,
general) solutions of the original differential equation.
The special Clairaut equation
y — xp — rVi + p1 = O,
when transformed hy the dilatation
takes the form y\ — x\p\ — o.
§§5 1, 5 2 CONTACT TRANSFORMATIONS 19 1
This simple differential equation has the obvious general solution y\ = cx\,
which is the equation of the family of straight lines through the origin. The
envelope of this family is the origin, which determines a union that is obviously
consistent with the relation defined by the differential equation. Passing back
to the original variables, the origin goes into the circle x- +/- = r'2, which equa
tion is therefore the singular solution'of the original differential equation, while
the lines through the origin go into the tangents to this circle. The equation
of their family,^ = ex — rVi -f c* = o, is the general solution.
52. First or Intermediary Integrals. — The differential equation
(136) 4>(x,y,p) = a,
for each value of the arbitrary constant #, has oo1 integral curves.
Allowing a to take successively all possible values, (136) determines
oo " curves which are the integral curves of the differential equation
of the second order
(•37) g = *. + ^ + *,J=<>.
The differential equation of the first order (136) is known as a first
or intermediary integral of (137). From the above it is seen that a
first integral of a differential equation of the second order classifies the
oo - integral CI/ITCS of the latter into ^families of oo1 curves each.
This classification is different, of course, for different first integrals,
of which there is an indefinite number. For
(138) *(x,y,p)=t
will also be a first integral of (137), if, and only if,
is the same as (137), i.e. provided
or
(121)
'p *X+lJW»
0.
1 92 THEORY OF DIFFERENTIAL EQUATIONS §52
Hence
THEOREM I. The necessary and sufficient condition tJiat
be first integrals of the same differential equation of the second order
is that <f> and \j/ be in involution (§ 49).
Starting with the function <£(.v, _v,/), a second function \j/(x, y, p)
will be in involution with it provided it satisfies the linear partial dif
ferential equation
(139)
This linear equation in three independent variables has two inde
pendent solutions, one of which is <t>(xtyyp). All of its solutions are
functions of these two. Hence
THEOREM II. Knowing <£(~r, v, /) = a, a first integral of a differ
ential equation of the second order, all of its first integrals may be
obtained by solving the linear equation (139). Having found a solu
tion of (139), independent of <j>, all the first integrals are given by
where <J> is an arbitrarily chosen function of $ and ty.
Since two independent first integrals
<£(.v, y, p)=a and i//(.v, v, /) = b
of a differential equation of the second order define the same set of
co2 integral curves but classified in distinct manners, for a particular
but arbitrary choice of a and />, say an and /v the differential equations
(140) <£(.¥, y, /) = tf(, and i/>(.v, v, /) = t>Q
will, in general, have an integral curve in common. At each point of
this curve both equations (140) determine the same value of/;
§52 CONTACT TRANSFORMATIONS 193
hence the equation of the curve
<o (x, y, 00, /;()) = o
may be obtained by eliminating/ from (140).
Still keeping tf0 fixed but allowing b to be an arbitrary constant,
the result of eliminating/ from <f> = #0 and \\i = b gives
CD (x, y, a0, b) = o,
a solution of <£(jc, y, /) = #0 containing an arbitrary constant which
is, therefore, its general solution. Hence,
THEOREM III. If a second differential equation
*<*JVJP>.-.<*
involving an arbitrary constant can be found sucli tJiat ff> and \\i are
in involution, the general solution of
4>(x,y,P) = a*
can be found by eliminating /> from the ftvo differential equations.
This process is frequently of service. (See El. Dif. Eq. §§ 25, 26).
Eliminating/ from (136) and (138) gives
w (x, y, a, b} = o,
a solution of (137) involving two arbitrary constants. It is there
fore the general solution. Hence,
THEOREM IV. If two independent first integrals of a differential
equation of t)ie second order can l>c found, its general solution is ob
tained I>\ eliminating p from tJic ecj nations of the first integrals.
Remark, — If <£ and \j/ are in involution, it follows at once from
the above that the two relations
determine an element union (§ 48) for all choices of the constants
a and b.
194 THEORY OF DIFFERENTIAL EQUATIONS §§52,53
It should be noted, however, that
(113) 0(^', v,/) = o, *t,(x,y,p) = o
may determine a union without the identical vanishing of [^>, \j/~\ •
thus see Ex. 6, § 48. But in every case when the relations (113) de
termine a union, [^, <£] must equal zero, either identically or because
of these relations. This follows readily from the fact that whenever
(113) determine a union, the equation of the curve of the latter is an
integral curve common to the two differential equations <£ = o, ^— o;
and conversely.
53. Differential Equation of the First Order Invariant under a
Group of Contact Transformations. The general type of differential
equation of the first order invariant under the group whose infinitesi
mal transformation is
is obtained (compare § 18) by equating to zero the general solution*
of the linear partial differential equation
On the other hand, the condition that the differential equation
/(•v> r5 /) = ° De invariant under the group whose infinitesimal
transformation is Bfis obviously ([12], § n)
Bf= o whenever f= o.
As was noted in § 51, a differential equation of the first order
(142) /=<o(.v,y)
arranges the oc2 lineal elements determined by it in oc1 unions, the
curves of which are its integral curves. If (142) is left unaltered by
*This solution is obviously the general expression for \\\e first differential invariants
of the group, the name given by Lie to invariant functions of x, y, p.
§53 CONTACT TRANSFORMATIONS 195
a contact transformation
(117) #, = X(x, r, /), yl = Y(x, v, />), /, = 7>(.v, r, /),
the latter interchanges the integral curves of (142) among them
selves, since it transforms unions into unions.
As far as the differential equation (142) is concerned, the only
lineal elements operated upon by (117) are those whose coordi
nates are (.v, r, / = o>(.v, r)). These elements are transformed into
(.v,, Vi, /, — o)(.v,, v,)) by (117), since the latter leaves the differ
ential equation unaltered. Hence the effect of the contact trans
formation (117) on the differential equation is the same as that of
the point transformation
(143) #! = X(x, y, w(x, y}), y\ = Y(x, y, «(#, y)\
Whence the
THEORKM. — If the differential equation
(142) f>=o>(x,y)
is invariant under tlie contact transformation
(117) a-, = A'(*, y, /), yi = y (x, y, /), pl = P(x, y, /),
it is also invariant under the point transformation
(143) x} = X(x, y, o>(.r, v)), y\ = Y(x, y, o>(.v, y)).
Both transformations interchange the integral curves <•>/"( 142).
It follows at once that if the differential equation
p = w(#, y)
is invariant under a group of contact transformations whose infinitesi
mal transformation is
Bf= «.v, r. / 1 V • | • ,,, .v. y, fy % + ,(.,; y, /) %
196 THEORY OF DIFFERENTIAL EQUATIONS §53
// is also invariant under the group of point transformations whose
infinitesimal transformation is
Uf= t(x, y, <r, v)) f£ + ?(*, y, «(*, y))*j-y-
Either of the methods of §§ 12 and 20, Chapter II, may then be em
ployed for solving the differential equation.
Remark. — Since BW=— WyW, it follows that the differential
equation
(144) W(xty,f) = <>
is invariant under the group of contact transformations whose infini
tesimal transformation has W^for characteristic function.
But the invariance is of a special kind. The effect of this infinitesi
mal transformation is to carry the point (x, )') of an element (x, y, p)
into (x + £ 8a-j y + 77 &z) where $ = IVP, TJ =plVp — W. The slope
of the line joining these points is
f -/--£=/ when UK- <k
I wp
Hence any element whose coordinates satisfy (144) has its point
carried in the direction of the line of the element, that is, the ele
ment and the one into which it is transformed are united (§ 48).
The infinitesimal transformation, therefore, leaves unaltered each of
the unions (§ 51) determined by the differential equation (144), and
the group has this effect on each of the integral curves of (144).
Such a group is said to be trivial with respect to the differential
equation (144), (§ 12), an^i is of no service in solving it.
APPENDIX
NOTE I
THE INFINITESIMAL TRANSFORMATION
In case both — (j>(x, y, a) and — $(x, }', a) vanish identically
da da
for the special value of a = a0, or if either of them becomes infinite
for that value of a, irrespective of the values of x and y that may
enter, a modification of the process for finding the infinitesimal trans
formation employed in § 2 must be made. It should be noted that
they cannot both vanish identically for all values. of a, for in that
case neither of the functions <£ and if/ could involve a at all ; nor can
either one of them become infinite for all values of x, y, ami a, since
<f> and i// are supposed to be generally analytic, which implies the
existence of finite derivatives, except perhaps for special values of
the arguments.
Let a be a value of the parameter for which — and -* are finite
da da
and at least one of them different from zero. The transformation
Ta determined by it has for inverse a definite transformation, 7^, of
the group, corresponding to the value « of the parameter, where a is
a function of a only. Since TzTa = Tn is the identical transforma
tion, TaTa+da is an infinitesimal transformation. If T* is
*\ = 4>(x,y, «)i J'i = $(*, y, «),
the infinitesimal transformation 7i7*a+sa may be written, when ex-
197
198 THEORY OF DIFFERENTIAL EQUATIONS
panded by Taylor's Theorem
x, = <f>(xlt ylt a + Sa) = x + <j>(xlt }\, u)8a* -\ ----
j, = >/'(>„ }\, a -f 8tf) = >• + — ^(xlf }\, a)8a -f •••,
since <£(.TI, ji, «) = ^, ^(xltylt a)—y. Owing to the way in which
a was chosen, neither of the coefficients of Ba is infinite for all values
of x and y, and one of them, at least, is not identically zero.
Writing
', a) , «] = ^(.r, ;', a) ,
,J'?«), «]^>;Gv,j, a),
(I45)
it follows that ^// infinitesimal transformation of the group (1} of the
type (2), § 2,
(2) £v = £8rt, 8>' = 17 80
r^« always be found. •
The forms for ^ and 77 found in § 2 are exactly what the above
become fcfr the special choice a = a = 0,,.
From the above it is seen that £ and rj in (2) depend upon the
choice of «. It remains to show how they depend upon* the choice
of the parameter. Let
8* = HO, j)&7, 8r = H(
or xl = x+ s,(x, .v)80, ji', = y 4- H (.v, j)8<7
be some known infinitesimal transformation of the group (i), where
E and II are not both identically zero, and neither of them is infinite,
in general. The result of performing successively any transformation
Ta of the group (i) and the above infinitesimal transformation is
Hcru J 0 (Vj.^'j, (/.) stands for
da
, a)
AIT KN I.) IX 199
some transformation of the group whose effect on the variables x
and y differs from that of Ta by an infinitesimal amount. In other
words, it is a transformation Ta+±a, where Art is an infinitesimal which
is a function of a and Srt only, because of the group property of (i).
From the first definition of this transformation
#2 = xi + »(!, }\)&a = <£O, )', « 4- H( <
while from*the second definition
x.z=<j>(x,yt rt + Art) =
' da
Hence
da
A,t,
Art
«(</>, i/OStf A*
(146)
for all values of :r, i1, rt and Srt, Art being a definite function of a and
Srt, and an infinitesimal along with Srt. By hypothesis H and H do
not both vanish identically; suppose, to fix the ideas, that E =£ o.
It follows that A* is not left unaltered by all the transformations of
the group ( r ) ; hence <f> must involve rt, and -* ^ o. With a proper
aa
choice of .v, v, rt the coefficient of 8a and that of Art in at least the
first of the two relations ( i.j(>) are different from xero. ]\y a theo
rem in the Theory of l-'unctions, concerning the inversion of power
series, Art is developable in powers of Srt, the development beginning
with the first power. Hence
r, A,/ is ri (unction of ,1 ;intl 8a only, the rorfticirnts in
',v iind an- tree ot r .ind v.
200 THEORY OF DIFFERENTIAL EQUATIONS
where w(a)^o. A<z is thus of the same order of infinitesimals as
8a. Putting this value in (146), dividing by 8a, and passing to the
limit &a = o,
(147) E(*,*)=7*(«)
or remembering that x = <b(x^ jr1} a), y = if/(xlt )\, a),
these may be written
Using (145), and replacing ^ and j', in these identities by x and y,
we have
(148) £(*,^,0) = _iE(*,j'), ^0,,fl) = _l_H(^^).
The effect, then, of using different values of the parameter in deter
mining an infinitesimal transformation by the method of the first
part of this note is to obtain pairs of coefficients of 8a in the two
formulae which are proportional, the factors of proportionality being
constants. Hence, by Remark i, § 2, all the infinitesimal transfor
mations so obtained are one and the same. We have thus arrived
at the
THEOREM. — Every one-parameter group of transformations
i
x\ = 4>(x, y, a), Vi = ^O, }', a)
has one and only one independent infinitesimal transformation
&x = £ ( xt y ) 8tf , By = rj( -v, v ) Sa,
where
.v v « »
da
APPENDIX 201
and <t is any value of the parameter such that at least one of f ^ \ and
[-JL] is not identically zero, and neither of them is infinite for all
\daja
values of x and y.
In general #0 is a possible value. In § 4 is shown that the trans
formations of the group can always be put in such form that this is
true. When for a given group this value cannot be used, this is due
to the way in which the parameter enters, and is not a peculiarity of
the group.
Remark. — This theorem and its proof hold for n variables without
any but obvious modifications to take account of the number of
variables.
NOTE II
SOLUTION OF THE RICCATI EQUATION
z^
air
In § 1 8 the general method for finding the differential equations
invariant under a given group led to the solution of the Riccati
equation
(39)
dx tdx t\df dx id
in which y, wherever it occurs, is supposed to have been replaced by
its value in terms of x and c [say y = <£(.v, '")] obtained from
u(x, y) = c, the solution of the differential equation
It is very easily seen that
(149) y
202 THEORY OF DIFFERENTIAL EQUATIONS
in which v is replaced, as above, by <£(.v, r), is a particular solution
ot (39). For differentiating (149)
dx £ \dx dy dx) ^ \dx dy dx)
Remembering that ^- = 3, this becomes
dx k
Whence follows at once that (149) satisfies (39).
It is a well-known fact that the knowledge of a particular solution
of a Riccati equation enables one to fiml a transformation of variables
which reduces the equation to a linear differential equation of the
first order, whose solution requires two quadratures (see EL Dif. Eq.
§ 73, i°). For the sake of simplicity, writing (39) in the form
and its particular solution • y' =y0't
the transformation / = - + yd
changes the differential equation into
£ + (Xl -f 2 yJXJz + X, = o,
which is linear. If z— w(.r, k) is the solution of this equation,
(150) / = __1_+>?1^(^)J
*(x,k) ^ t[.x, +(x, c)]
is the solution of (39). Solving (150) for k, and replacing <£(.r, c)
by->'' «'(x,y,S)=*
is the required second solution of (37), § 18.
AITKXDIX 203
^=x*= 5
— y x I +y*
tf.,i j _|_ v-
The Kiccati equation (39) is - • . - = -•
ax _ vV —
The transformation
s _v
dz . -zx I
reduces this to • - 2 = - —
^r c — x~ Vf — x-
— " . 7 /
Integrating, - + k{c —
I lence / = . ^T^T^T ~ ~ '
and k = (TT^TT^ -u ^ ^ W' ^' 7' ^'^ '
Compare this with II, § 19.
NOTE III
ISOTHERMAL CURVES
The condition that two distinct families of curves
<£(.v, j) = const, and i^(,v, y) = rrv/y/.
divide the plane into infinitesimal squares may be obtained from the
following considerations : *
Passing to the new system of coordinates
the two families of curves have the simple equations
x = cons/, and y = const,
*AH this holds, practically without change, for isothermal curves on surfaces.
204 THEORY OF DIFFERENTIAL EQUATIONS
From dx = ^ dx + 3* dy and dy - * dx + d* dy,
dx dy " ox dy
dy dy dx dx
A+>
-here
the Jacobian of <£ and ^, which is not identically zero, since the two
families of curves are distinct. (See El^ Dif. Eq. Note I of the
Appendix.)
The expression for the element of length of arc of any curve in the
plane, in terms of the new coordinates, is
d? = do? + <if= Edx*-2Fdx dy + G dft
where the coefficients
dy dxdxdydy G = \dxj \dy
U ($>
are to be expressed in terms of x and y by aid of (151).
A first requirement, that the two families of curves form isothermal
systems, is that they cut each other orthogonally. The condition for
this is 1,1,
d<£ d\l/
dx dy
-T4> = ty'orF=°-
dy dx
Hence a necessary condition is that the expression for the element
of length of arc assume the form
(/s- = E dx1 +
APPENDIX 205
For a curve of the family x = const, (which will be referred to as an
jr-curve)
dsx = ^J G dy,
while for a y-curve ds = "y£ dx.
If -\/~E = V£ at every point in the plane, the curves divide the
plane into infinitesimal squares, for choosing dx the same as dy,
Moreover, if V ' E and V G contain a common factor, and each of
the remaining factors is a function of the corresponding variable
only, thus
AOr, »)/?(,),
the introduction of the new variables
X=Ca(x)itx and Y= Cft(y)tfy
gives </jT=A(Ar, Y)JXandJsT = A(X, Y)dY9
where A(X, Y) is what \(x, y) becomes when x and y are replaced
by their values in terms of X and K The families
X= const, and Y= const.
(which are obviously the same as x= const. and y — const) have the
desired property. Hence the
THEOREM. — The necessary and sufficient condition that tlie entires
<f>(x, _)') = const, and thrir orthogonal trajectories i//(.v, \}— const.
divide the plane into infinite si inal squares is tJiat tJie choice of variables
reduces the expression for the element of length of arc to the form
,K- = X2(jr, y} ] [«(jr)//r]2 + [0(y ^]2f ,
where, in particular, <c(x) and J3(y} tnav eacli he unity.
2O6 THEORY OF DIFFERENTIAL EQUATIONS
Thus in the case of a family of concentric circles and their orthogonal trajec
tories,
x = x2+y*, y--,
2x
Putting X = log Vjr, Y — tan'1!/,
For other examples of isothermal systems, see § 24.
NOTE IV
DIFFERENTIAL EQUATION OF THE SECOND ORDER NOT
INVARIANT UNDER ANY GROUP
If the differential equation be written in the form
the condition that it be left unaltered by the group
( 6 1 ' ) — £ rj - — T/— - + 17" = o, whenever /' = F (x, y, / ) ,
t(5S),ia63,
~-
Al't'KMtlX
207
Replacing y" t wherever it occurs in y", by F(x,y, y') the condi
tion (61') becomes
fr?_ _ _ -.
dy 2 **
-
dx'By
ty W
-v
dy
for all values of x, y, y' .
Since (152) is an identity with respect to x, y, and y', it is equiva
lent to a number of differential and finite equations in £ and ry, the
exact numl^er depending on the form of F. Fixing one's attention
ony alone, (152) is equivalent to at least four equations, and per
haps more. In general it is impossible to find functions £(x, y) and
ij(x> }') to satisfy all these conditions.
As an example, consider the differential equation
The identity (152) leads to
tan/.
dy
= 0,
the dots standing for terms free of tan y' and sec y and involving
second derivatives of £ and tj. (See below.) This identity implies
the following relations :
208 THEORY OF DIFFERENTIAL EQUATIONS
«!=<>•
From (#) and (/) -~ = 3- - = o.
These together with (V) and (</) make it necessary that
£ = const, and 7; = <r0«j/.
Hence, the omission of terms involving higher derivatives of £ and
rj above.
Since (6) must hold for all values of x and y
£ = r) = o;
i.e. there is no infinitesimal transformation and, therefore, no group
that leaves the differential equation unaltered.
Remark. — The case of a differential equation of the first order is
entirely different. The condition that
be invariant under 6/= £ ,. + 77 J-
ox ' oy
may be put into the form
Here one of the functions, say £, may be chosen at random, leav
ing a partial differential equation in T;, which always has a solution ; as
a matter of fact it has an indefinite number of them. This is in
entire accord with the result arrived at in § 15.
APPENDIX 209
NOTE V
The symbol of the infinitesimal transformation of the extended
group corresponding to 6^= £ 7) ~^~ ^ ;} ^s C(24)> § T3]
dx By\dx dy
Introducing the symbol
£//"may be written in the form
and (BU')f = J}(Jl/+ P(X, y, /) ,
where p(x,y,y')= B(By —yfJB£) is some function of x,y, y', whose
actual form is of no importance in this discussion. Introducing the
additional symbol
be written in the form
Also (CU)f=<r(xtyt]f)Cf9
where cr(.v, v, r') = C(fty — y'B£) is also a function of .v, r, r', whose
form is of no importance here. The fact to be emphasized is that
{Blf)f and ( Ct?}/ arc linear functions of Bf and Cf, the coefficients
being functions of x, y, y' .
210 THEORY OF DIFFERENTIAL EQUATIONS
Moreover, if U^f and U.>f are any two groups, (/?(£//£/,'))/ and
(C(C/i &>))/ are also linear in Bf and Cf. For from Jacobi's
identity (§ 36)
(B( U{ U{) )/+ ( U{ ( MB) )/+ ( Ui (BUI) )/ = o,
and in an analogous manner,
( C( U{ W) )/ = ( U^, - U^ Cf.
Since (U^U.^f is of the same type as Uf (§ 14), it may be written
and
Noting that (U{ U*)f coincides with (U^)'/ in the first two
terms, at least, we may put
It remains to show that o> = Brj
The alternant of /?/and ((SiC7»)/is
This being linear in /y/and Cf, as was proved above,
.'. Bt = A, />'ry - <u = A/.
.\ITK\ni\ 211
Whence co = ^-/^.
This, as was noted above, establishes the identity
Remark. — It can also be proved that for the ;«-times extended
groups ^""
NOTE VI
CONTINUOUS GROUPS INVOLVING MORE THAN ONE
PARAMETER
r-parameter Group of Transformations. — The aggregate of all the
transformations *
x1= 4>(x,y, 0i, tfjj, •••, ar),
(153)
h'l = t('X>}'> al> a* •", <*r)>
obtained by assigning to the parameters alt a.2, •••, aT all possible
values constitutes a group, if the transformation resulting from the
successive performance of any two of them is one of the transforma
tions of the aggregate.
As in the case of one-parameter groups (Chapter I), the groups
here considered arc supposed to have their transformations pair off
into mutually inrrrse ones. That is, corresponding to -any set of
values of alt a.>, •••, <rr there must always be another set <r}, <7.,, ••-, ar,
* As before, 0 and <// ;ir»- supposed (o he <M'ner;illv analytic real functions of x,y, alt
a-2, •••, <ir : and, unless specially stated, it \\ill be understood that .r and r are real
and tliat the parameters take sueh values only as render .r, and v, real. Groups of
transt'oi inations involving two variables aie considered here. For the theory of those
involving ;/ variables the student is referred to Lie's works, especially his 'rraiisforma-
tio*jgrttppettt Vol. I., and liis Coiiti/iuit-rUc/ic ( if u />/><• a ; also to Campbell's Introduc
tory I > <\itis<- on /./V.v Theory.
212 THEORY OF DIFFERENTIAL EQUATIONS
(functions of the former ones) such that
Another way of putting this is: If the equations (153) are solved
for x and 7, the latter must appear as the same functions, <£ and \f/}
respectively of x± andj^ and a set of r functions of alt a2, • •-, ar) as
indicated by (153).
Thus, consider the translations
XVII xi = x + ai, yi - y + a2.
If one of them be followed by a second one,
x-2 = *i + b\y y-i =y\ + bzt
the result is x% = x -f- c\, y« ---- y + c%,
where ^i = #1 -f b\, c-i =
Solving the equations XVII for x and y,
x = x\ — « i, y — y\ —
Hence a\ =— a\, a-i =— <*%•
Again, consider the displacements
XVIII x\ — .rcos a\ — jsinai + a»* y\ —
A second transformation of this type
x.2 = x\ cos bi — y\ sin l>\ + b^ y-i — x\ sin b\ + yi cos b\ + ^3,
results in
x-2 — x cos ri — y sin o + ro, jo = jr sin c\ + ^ cos <TI + ^3?
where
f! = rfi + ^if r2 = a-2 cos ^i — a;5 sin 61 + b^ CA = a» sin b\ -f a?, cos ^ -f
Solving the equations XVIII for x and y,
x = ^icos(— tfi) — /isin(—
j/ = jfi sin (— tfi) + Ji cos(—
Hence
i + rt;j sin </i), «;$ = (J-i sin <?i — a-\ cos «i.
APPENDIX
213
In the case of the general projective transformations
4 a±y 4 a* — a*x 4 <*& v 4
— , y\ —
&1* 4- a%y + #9
there is no difficulty in seeing that these constitute a group. For if one of the
above transformations be followed by
-f
there results
where
= «* + C^ + <* , y, = <** + <*? +
^7-^ + ^8>' + ^9 ^7-^ + C*y +
c\ — a\b\
4- <?7^3,
= a-2t>± 4- rt6*5 4 «8^6,
— a36f 4 «o^s 4 <? 9^8,
^"9 — «r,^7 + #6^8 4 ^9^9.
Moreover, the result of solving the equations XIX for x and y is
4-
where /^i, /4o,
determinant
4 ^e^i 4- ^9 ' ^3^1 4- ^c^i 4 -
are the cofactors of the corresponding elements of the
a\ 0o
a^ «6
Since the successive performance of two mutually inverse trans
formations results in the identical transformation, the latter must
always be a member of the Lie group ; hence there must always
214 THEORY OF DIFFERENTIAL EQUATIONS
i
exist a set of values of the parameters, a", a.?, ••-, <zr°, such that
(I53o)
It is readily seen that for
XVII, ai0 = aj = o;
XVIII, -tflo = «2o = fl.,0 = o ;
XIX, «i° = «5<> = «9° = tfwy number (different from zero},
aj = tf3'> = «4° = rt6° = <Z7° = tf8° = O.
We shall further presuppose that all of the /'parameters in (153)
are essential, that is, that the formuke of transformation cannot be
replaced by another set involving a smaller number of parameters
without reducing the number of transformations represented by
them.
Thus xi — x -f ai + a3, yi —y + a->
contains no transformation that is not included in XVII. It involves only two
essential parameters ; a\_ -\- a?> is no more general than a\.
In XIX, as is well known, there are only eight essential parameters ; since
the expressions are homogeneous and of degree /ero in the parameters, it is only
the ratios of the latter to any one of them that count.
A group involving r essential parameters is known as an
r-paramcter group.
It is frequently possible to tell by inspection whether the parameters
appearing are essential or not. An analytic criterion is given by the
theorem of Note VII.
Show that the following sets of transformations constitute groups.
Find the respective values of the parameters that give the inverse
and those that give the identical transformations :
Kx. i. a-j = a^x + a.2t )\ = al y + a*
Ex. 2. ,rj = a& + azy + a3t y\ = a^c + <75 y + a6.
Ex. 3. ji\ = x + air + tfj, y\ =y + «3.
APPENDIX 215
Ex. 4. xl = x + rt,jr + a.>, yi = <r .. v.
Ex. 5. ^ = (#! -|- i ),v + O, —
Infinitesimal Transformation. — The transformation
y, = f(x, y, < + 8alt of + &»2, -n tfr° + &*,),
where ^^ f72°, ••-, «rn determine the identical transformation and
fail, &*2, "•> S^r are infinitesimals, changes jc and y by infinitesimal
amounts, since <£ and ^ are supposed to be continuous functions.
Developing by Taylor's Theorem,
and stand for what
da?
respectively become when al = af, a.2 = a", •••, ar = a°, and the
unexpressed terms are of higher degree than the first in 80lf Sa.2, •••,
Bar. The changes in x and ^ are then
2l6 THEORY OF DIFFERENTIAL EQUATIONS
We shall suppose that at least one member of the pair
,
for each value of / from i to r, does not vanish identically, and that
all of them are finite. Calling them &(#,.}')* anc^ 'ft (•*».)') respec
tively, the transformation may be written
In exactly the same way as is done for one-parameter groups in
Note I, it can be shown that infinitesimal transformations of the form
(155) always exist, even when the parameters enter in such a way that
for the particular values a®, a.2°, • •., ar° both members of some pair
3«K*, y, *°) dt(x,y, a0)
da? da?
vanish identically, or if some one of them becomes infinite. f
Here &/b &a2, •••,8<zr are any infinitesimal increments of the first
order. Taking &a as a standard infinitesimal of the first order, we
may put
* Here ^ and 77,- are written as functions of x and y only, since Oj0, a.f1, •••, ar°
appear as numerical constants.
f The general expressions for the coefficients in (155) are, in thenotaiion of Note I,
Here «lf «._,, '••, «r are any set of values of the parameters for which both t. and 77*
are finite, and at least one of them is not identically zero. The forms (155) for £,, 77,-
are what the general forms' (155') become for the special choice ak = (ik = at0
(*= i, 2, ...,r).
APPENDIX 217
where tlt e2, ••-, er are any finite constants. The general type of an
infinitesimal transformation may then be written
^ 6x
v = (/IT?! 4- ^r?,
For the sake of brevity we shall write
(157) &x = £Sti+ •••, 8v = i/80 + — ,
where £ = *<^ an(^ ^ = ^*" Intr°ducing the symbol
and similarly
we have
(158)
It can be proved * that when the r parameters of the group are
essential Uif, U^f,-- Urf} (in which £., ?;„ are given by (155') for
any properly selected set of values of the parameters, in particular
they may have the special forms (155)), are linearly independent ;
that is, that it is impossible to find a set of constants cly c.,, •••, cr such
that c&f+ cMf+ '-+ crUrf= o,
which is equivalent to saying that for no set of constants r,, r,, •••, cr
can both the relations ,.
* Thus, for example, see Lie, ContinuierlicheGruppen, Chapter 6, or his Trans form a-
tioHsgruppen, Chapter 4. Also Campbell, loc. cit., § 42.
The object of this Note is to present as compactly as possible, consistent with a
clear understanding of the chain of reasoning, the relations luturrn /--parameter
groups and their infinitesimal transformations. Consequently when long and tedious,
the proofs of certain facts are omitted here. These may, however, be obtained from
the references given.
218 THEORY OF DIFFERENTIAL EQUATIONS
hold simultaneously. Moreover,* it can also be proved that if
&v = E8rt+--., 8_v = H80 + •..,
is any infinitesimal transformation of the group, H is a linear function
of £lf £2, • ", £r with constant coefficients, and H is the same function
Of r,,, T?.,, • • -, ,,„, thllS,
where the set UJ= L + rli (/ = i, 2, •••, r)
dv c^
is any linearly independent one.
The coefficients of &z in (156) can therefore never both vanish
identically. Hence at least one of the terms of first order must
appear. Infinitesimals of higher order than the first may conse
quently be neglected, and the infinitesimal transformation may be
written in the form
(159) &* = (f& + t& + ••• + f&) Stf, 8v = (^,17, + e,^, + ... -f erri,)&a.
The change in any function f(^,v} produced by (159) is then
S/EEE Vf*a,
where
(158) 0^4£%/+.'.l%/+ - + ^r^r/
as in § 3, is the symbol of the transformation (159) and will be usrd
to represent it.
The above may now be expressed as follows :
THKORKM !.• — AVvvv continuous Lie ^r<>///> intNthring r essential
parameters contains r linearly ini/epeiittenf infinitesimal transforma
tions (\/\ t\,f, •••, frrf, in terms of which every infinitesimal trans^
f(>rmalion of tJie i^nn/p can be e.\ pressed linearly until eons taut
coefficients, tints
(158) uf^c,i\f+c.2r,f + - +crurf.
J/"/'/vv'7'r/-, fi'frv transformation of flie type (i^.for all" choices of
///e <'on slants eVi <"._,, ••• <",., l>elt>ugs /<> (lie group.
APPENDIX
219
Remark i. — It follows that in any set of infinitesimal transforma
tions of the group, only /• at most can be linearly independent.
Moreover, starting with any r linearly independent transformations
UJ, UJ, •-, UJ, every set
VJ = *u 1 1/+ <•* UJ + - - - + ^ UJ
(k= I, 2, •••, /')
will be linearly independent provided
=£0.
Any set of r linearly independent infinitesimal transformations,
V\fi V-2/> '"» Vr/> may ^)e taken as the r transformations (referred to
in Theorem I) in terms of which all the infinitesimal transformations
can be expressed linearly with constant coefficients ; for, since A ^fc o,
each of £/,/, UJ, •••, UJ is a linear function of /',/, VJ, ••>, Vrf
with constant coefficients.
In the case of XVII
A set of linearly independent transformations is
U\ f— f (.'•/'
~~ dx*
In the case of XVIII,
A set of linearly independent transformations is
-f Sa-.\ =
220 THEORY OF DIFFERENTIAL EQUATIONS
In the case of XIX there are only eight essential parameters. Putting ag = I,
r _ „ i *r _ (i + 8<*i)x + y 3*2 + da3
X\ — X -f- OX — - — •
x da7 + y 5« 8 -+- I
But - = I — x $a-, — y da$ + • • •,
where the dots stand for terms of higher degree than the first.
... x\ = x 4- 8x = x -\- x?>a\ + y
Whence, 5x = (e±x + e»y + €••>, — e-jx2 — e%xy}ba.
Similarly, 8y — (e±x -\- e^y^- e$ — e-^xy — e8}>'2)8a.
A set of linearly independent transformations is
Ex. 7. Find the infinitesimal transformations of the groups in
Ex. i, 2, 3, 4, 5, 6 above.
Group Generated by Infinitesimal Transformations. — Starting with
the infinitesimal transformation
(158) 67 = ^7+^7+ ... +eruj
in which the constants elt e^ ••«, er are fixed, the finite transforma
tions of the group generated by it may be obtained either by finding
those solutions of
(i 60)
APPENDIX 221
for which xl = x and yl = y when / = o (§ 4), or in the form (§ 5),
£
(161)
In both cases / is the parameter, and / = o gives the identical
transformation.
If elt e2, •••, er are arbitrary constants and c^/, £/,/, •••, U/ are
linearly independent, the infinitesimal transformation contains r— i
parameters (viz. the ratios of any r— i of the <?'s to the remaining
one), and the general expression (161) for the finite transformations
generated by it contains r parameters. That these parameters are
essential follows from the linear independence of £/,/, c7,/, •••, £//.
A proof of this fact may be found in Lie's Continuierliche Gruppcn,
pp. 186-190. Hence there are ocr transformations in the set (161).
If £/i/", U%f, -", 6//are r linearly independent infinitesimal trans
formations of an r-parameter group, every transformation of the set
(158) belongs to the group (Theorem I). All the transformations
of the one-parameter group generated by any transformation (158)
belong to the r-parameter group (Lie, Continuierliche Gruppcn,
p. 183). The ocr' transformations (161) therefore belong to the
group. Moreover, every transformation of the r-parameter group
(at least all such for which the values of the parameters are suffici
ently small so that when developed by Taylor's Theorem in powers
of the parameters, as (161) are, the series are convergent) is in
cluded in (161) (Lie, Transformati0nsgrupp€itt Vol. I, Ch. 4, § 18).
Hence
THEOREM II. If " U\f, U»f, •••, UJare r linearly independent trans
formations of an r-para meter group, the latter* is precisely the OggTV-
* At least all its transformations corresponding to values of the parameters which
differ by limited amounts perhaps (see above) from those which give the identical
transformation.
222 THEORY OF DIFFERENTIAL EQUATIONS
gaff <>/ all the one-parameter groups generated by the cc7—1 infinitesimal
transformations
Remark 2. Since / and the <^'s appear in (161) in the combina
tions /<?!, /if2, •••, ter, there will be no loss in suppressing the / and
writing the finite transformations of the group in the form
(161')
I
2~!
where the <r's are now r distinct parameters. The identical trans
formation is given by el = e.2= •»=er = o, and the inverse trans
formation by ei ; = — ci (/=i, 2, •••, r).
In the case of XVII the general type of infinitesimal transformation is
The finite transformations (161') are seen at once to be
Xi = X + «?!, jj'i =^ -f t».
In the case of XVIII
.'. #i = x — o^ 4- ^ - -1- e{-x + -1- epy + --*!4jr — •••
= x cos <TI — y sin <? i -f *••_>.
Similarly ^'! = JT sin e\ -f- ;' cos <TI + ^3.
Remark j. The expressions for jr, and r, in (161') may at times
become extremely complicated, as for example in the case of the
group XIX. Also the actual problem of integrating equations (160)
with the r's arbitrary constants is usually a difficult one. To over
come this practical difficulty T,ie suggested the following method,
which was also given independently by Maurer ( Matli. ./////., Vol. 39) :
APPENDIX 223
Having found
(162) *}. = 4>i(x, y, <O» yi = <l>i(x>y>ai)>
(/ = i, 2,..-, r),
the finite transformations of the one-parameter groups generated by
each of the r linearly independent infinitesimal transformations
U\ft U.>f, -•-, £7r/of an /--parameter group, the result of performing
successively one transformation (with arbitrarily selected parameter)
out of each of the ;• groups (162) is a transformation belonging to
the r-parameter group and involving the r parameters alt a.,, •••, ar.
That these are essential follows also from the linear independence of
t Urf. (See Lie, Continuicrlichc Grnppcn, p. 194.)
In the case of XVII
=&
flbr
U*f= (f : u-g = xlt
By
The successive performance of these gives
*z = x 4- rti, y» = y
In XVIII
rr f <V , 3/
t/i/= —y -^ 4- •«•— : -*i =
=- =
The successive performance of these gives
jc-.\ — A cos <zi — _j' sin </i + </;., _j';! = x sin ^/i + 7 cos ^i 4- r/:;.
In XIX
U\f=(. : xi=x + rt|, Vi = j-, ,
224 THEORY OF DIFFERENTIAL EQUATIONS
~dy '
df
Usf=y £ : x* = xb, y& = ^5,
dx dy i - «8j'r
The successive performance of these gives
x _
_
where <4=
Find the finite transformations generated by the following
Ex. 8. U=
oy
Ex. 9. « * + ** + *i^ + +'t
Ex.10. &&(<& + * + €.
ox oy
Ex. 11. Uf= (etf + ^2) I"- + 'V'f .
3jc 61)'
Ex. 12. £/EE (^ 4- e«x H- <, v) ;X + (',
(U
Ex. 13. ^- (^ + ^2^ 4- W) 4
APPENDIX 225
Lie's Principal Theorem. — It was shown above (Theorem II)
that if £/!/, £/,/, ..., UTf are r linearly independent infinitesimal
transformations of an r-parameter group, the aggregate of the cor
transformations of the ccr~l one-parameter groups, each generated
by an infinitesimal transformation of the set
(158) Uf = f \Uif-\- e^U-J -\- •" + eTUrf,
forms an r-parameter group. On the other hand, starting with any
r linearly independent infinitesimal transformations L\f, U.J. •••,
Urf (without knowing whether they form a complete set for some
group), there is no reason to suppose that the oor transformations
generated by the various transformations (158) form a group.
Thus, starting with £/I/=^T, EV=*f£'
CM vy
the transformations generated by
_-.- of of
LJf =: e\ — — \- f-~>x -7—
dx dy
are xi = x -f a\, y\ = a»x + y + et-^-
While these transformations involve two essential parameters, it is very easily
seen that they do not form a group.
A definite answer as to when the oor transformations generated by
the various transformations of the set (158) form a group is given by
LIE'S PRINCIPAL THEORKM : * The necessary and sufficient conditions
that the oor transformations generated by the ocr~1 infinitesimal trans
formations TJ f +- TJ / -U -4- Tr f
* Lie calls this theorem " Der Hauptsati der Gruppentheoric," and gives a proof
of it for groups involving two variables in his (\wtinuifrlichf Grtippen, Ch. 12. In
his treatment of the general theory of continuous groups, this theorem is the second of
his "three fundamental theorems." See his (\>ntinuierliche C,ruf>f>en,(Z\\. 15, or his
Transformation wupp.'n. Vol. I, Ch. 9 ; also Campbell, /,»r. cif., Ch. IV.
A detailed proof of this theorem would be beyond ihe scope of this Note. A state
ment of it with illustrative examples will suffice.
Lie first deduced this theorem in 1874.
Q
226 THEORY OF DIFFERENTIAL EQUATIONS
where U^f, U-,f, •", UJ are linearly independent and <?lf e.2, • ••, er are
any constants, constitute, a Lie r-parametcr group arc t/iat
(/,£=i, 2, ..-, r),
where the c's are constants.
Remark 4. — This theorem is equivalent to the following two :
i°. TJie infinitesimal transformations of an r-parameter group form
tin r-parameter group of infinitesimal transformations. (§ 43.)
2°. 27u' transfoi-mations if tlic groups generated b\ tJie transforma
tions of an r-para meter group of injin itcsimal transformations form an
r-parameter group.
In the case of XVIII,
Here (l\U*)f=Uzf, (U,U^f~-U^ft (U,U*)f=o.
In XIX
( ^, £/7y= 2
and so on.
Ex. 14. Show that the infinitesimal transformations in Ex. 8 to
13 satisfy the conditions (163).
NOTE VII
CONDITION FOR ESSENTIAL PARAMETERS
The r parameters in
(J53) -vl = <#>(.v,.v, a}, a.,, ••-, ar), )'1 = t//(.v, r, alt a,, --, ar)
are //^/essential if (153) <~an be n-nlared by
(164) j;1 = *(jr,^, «!, «,, ..-, «r_TO), .Vi = ^Cv,.v, «i, «,., •••, «r_m)
APPENDIX 22/
In this case the identities
(165) <£ = <!>, ^ = *
for all values of x and y, determine alt a2, •••, ar_m as functions of
a\> a-2, -~, (iT ; f°r by saying that (164) replaces (153) is meant that
as soon as the #'s are given a set of the a's is determined (not neces
sarily uniquely) which will give rise to the same transformation.
A homogeneous linear partial differential equation of the first order
in r variables
(166) 4/~Xl(alf a,, •••, ar)---f--\ ----- hXrOi> «z, —j <*r)~- = o
oa1 dar
is determined uniquely by r— i independent solutions.* An equa
tion of this type can therefore be constructed which shall have for
solutions
«!, «,, •••, «r_m, /?r_m+1, •••, (3r „
where /3f_m+l, •••, /?r_], any convenient functions of the #'s inde
pendent of the «'s, are added to the latter to make up the number
r— i in case ;// > i. This equation will have for solution also any
functions of the «'s, in particular $ and >P, x and y appearing as
parameters ; or owing to the identities (165), by which the «'s are
defined, <f> and ^ will also be solutions.
Conversely, if <£ and \j/ satisfy an equation of the type (166), they
are functions of some or all of its r — i solutions.
i.e. the #'s enter <#> and \f/ in such a \vnv that for all values of A* and y
<£(.v, v, <7., t7... •••, ^7,) =.1 ^(.v, r, y,, y,. •••, yr_0,
^(.v, i', ^,. <A_,, •••, ^7,) = M'( v. r, y,, y,,, •••, y,^).
* A proof of this for the rasr of r 3 is i^ivcn in tlu' in -t footnote of ij 34. The
proof for r any number is exactly the same.
228 THEORY OF DIFFERENTIAL EQUATIONS
Hence the
THEOREM. — The necessarv and sufficient condition that the r para
meters in (153) be essential is the impossibility of Jin ding r functions
of them xu Xt> '"•> Xr Sltc'h that the resulting linear equation (166)
shall have <£ and \\i for solutions.
Remark. — There is nothing in the above to show whether the
r— i parameters y1? y2, •••, yr_: are essential or not. The same test
must be applied to them also, unless, as is frequently the case, the
exact state of affairs is obvious on inspection.
To illustrate, consider the transformation
xi = xa]°sb -f #<*« -1- c = <}>(x, y, a, b, c),
yi =jj/«log6 = \f/(x, y, a, b, c}.
If a, l>, c are not essential it must be possible to find three functions of them,
Xi(fl, b, f), X2(X b, c), X$(a, b, <r), such tliat the equation
(i 66) Af=XM + x>M+x,V=o
da db Qc
is satisfied identically (for all values of x and jy) by 0 and ^; that is
,=1^(^108* +
a
\ a b
for all values of x and y. These two identities are equivalent to
= o,
= o.
APPENDIX 229
By inspection, a set of forms for xi» X2» %3 are found to be
Hence the three parameters are not essential.
To express the formulae of transformation in terms of a smaller number, one
proceeds to solve the equation
(166') a\oga&- b\ogt>i¥=o. *
da fib
Passing to the corresponding system of ordinary differential equations
da db _ dc^
a log a — b log b o
it is obvious that log a log b and c
are a set of solutions of (166'). Putting
log a log b — «, whence
the formulae of transformation take the form
or, more simply still, ^i = a^ -f
TABLE I
IN this table is given a list of the more readily recognizable forms*
of differential equations of the first order which are known to be in
variant under certain groups. The same type of equation is some
times given in various forms, and special cases are also noted when
this seems desirable.
In the second column appear the groups under which the equations
are invariant. The numbers are those employed in § 19. For the
sake of simplicity / and q are used instead of - and J. respectively.
dx oy
The corresponding integrating factors of § 12 are given in the
third column.
In the fourth column appear the canonical variables. f
When variables which are separable in the transformed equation
(§ 20) can be obtained easily, they are given in the fifth column;
the form of the group resulting from the introduction of these vari
ables is given in the last column.
* Other forms will be found in $ 19.
t Then- is :i certain dearer of freedom in the choice of cnnonie;il variables, since
they are particular solutions of the differential equations (i^'l. ^ 10, or of the eorrcspond-
d/'
Ing ones in case the group is to be reduced to the form , • Moreover, the right-hand
member, i, in one of these equations may be replaced by any convenient constant (see
Remark i, $ 2) ; use of this fact is made when it will simplify the form of the resulting
variable.
231
232
THEORY OF DIFFERENTIAL EQUATIONS
HI
* £
^ a
^a
u -
52
-f
I
-
o" w
ii §
o
II
/-~s
X
o
"V
V
TABLE I
233
ANO
ARI
II II
-
t
X
"\ 8 v.
•ft1*
g " "
*& *
</i •" ti
3 £ c
s
J- ? >s
+ !r +
>
^ B. *S
+ C +
234
THEORY OF DIFFERENTIAL EQUATIONS
« -f
^
EQ
•2 ^
rt .
^J
^
.
_
«j a^ "3 1
+ 1
TABLE I
235
-H
-H -i
H *
i a
'H 0 C
:
.
i ,2 H ^
K-
H-
. — ••
1 1 -e-
r
o
II
^
+
?
»*\
-H
-H
; s
i i
^ ^
i i
TABLE II
General types of differential equations of higher order than the
first, invariant under given groups, are usually complicated and not
easy to recognize. In this table are given a few which a little ex
perience will enable one to recognize.* Such characterizations as
are simple are added.
DIFFERENTIAL EQUATION
GROUP
CHARACTERIZATION
/(*,/,/', -,yW) = o
i; q
y is absent
/(y,y',y",-,y^=o
i';/
.r is absent
f(ax + by,y',y", ... , ;/(O) =O
XII; bp- aq
x and jy enter in the combi
nation ax -f by only
iiii-fh
HI; yq
Homogeneous my,y', •••, y(r)
f(y y> y^\ o
VI ; xp -f- nyq
Homogeneous \\hen weight!
of x, y, y1, •--, \>(r) an- ', n,
'U»' *«-«' ' xn-r)
n — I, ••• , n — r respectively
/K /.*/',-,
\x .
xr-iy(r) J=0
IV; xp +yq
Special case of above, for n — i
f(y, xy', x2y", ... , x*y(r))=o
III'; xp
Another special case, for n = o
f(x,<t>y'-<t>'y,<t>y"-<t>"y, ...,
<f>y(r) — $(r}y} = Q
VII; 0O)?
A linear function of the vari
ous elements, except x, gives
rise to a linear differential
equation
ftxtxy'-*y,*?y"-*(k-i)y,
••-, xry(r)—k(k-i)...
(k-r+i)y~\=o
VII; ^
A special case, for 0(.*) = x&
* Other forms will be found in § 28.
TABLE II
237
DIFFERENTIAL EQUATION
GROUP
CHARACTERIZATION
f(*,*y'-y. /',/",•••,
gy(r))=o
VII; xq
A more special case, fur
0O) = *
/ y y"y s—l\
/('v-"'> + 7-)=0
VII'; ysp
•
/H-^H
VII'; j/
/(*,.>y,;'/'+/2)=o
VIII; ^
/fe */-;•» *y') = o
X ; ^-2/ + xyq
/Oj, *•/ +,v, xy" + 2/)= o
X;/-^y
,fr,,,2 izs! ym \
II; -^/+^
Each of the elements appear
ing in the differential equa
tion has a geometrical signifi
cance, which assures invari-
ance under the group of rota
tions
'V l}' x+yy» (i+/2)3J
= o, or
f(#++ *+yy' y"'2 \
f\ l}' Vi+y^'Cu-y^J
fa+Aj^r-j!*^)
\ Vi+/^ '(1+y"?)
= 0
ANSWERS
Section 1
1. a = - ; ao = I ; the equilateral hyperbolas x\y\ — xy = const.
a
2. a = ; dQ = i ', the parabolas •?— =• ^— — const.
3. a = _ ; fl0 = i j the semicubical parabolas *-*- = «— = const.
(i x\^ x^
4. tf = — « ; rto = o ; the ellipses x\* -\- 2y\* = x'* + 2^v2 =: const.
5. # = — « ; ao = o ; the equilateral hyperbolas xfi — y\* = x^ — y* = const.
)'i v
6. a = — a • ^ = o ; the straight lines — ~ = const.
7- a = — 'y ao = I ; the straight lines ^i = jy = const.
8. fl = — a ; ao = o ; the spirals log \/.rr + J'i" — tan-1 ^1
= log vV2 + y2 — tan-1 £ = ^«j/,
Section 2
•*' 2>/ 7- f =
5. |=;', >n = x.
Section 4
4. x? - .v- + 2 /, ^!2 =;'2 - /. .-.*!=+ Va-^ + 2 /, j/i = + Vy2 — /.
5- -VL +jJ'i = ^
or
.n = JT cosh t + y sinh /, ;'t = x sinh / + v cosh /.
6. », -=_,. ... ^=_^L_, ^l = ^L_.
.^i x •> i Jf I — .v/ ' I - .r/
7- .''i =J, .r, +^i = **(jr + V). .'. xi-f*x-\r(et— l).r, v, =jj/.
8. .vrl.rr <'-'(.\- ' i-'-', tun ' -1'1 = tan"1 ^ -f- /. .'. :ri = «f(.vct>s/— j-sin /),
240 ANSWERS
Section 5
2. xi = e*x, y\ — (fy. 3. XY - e-'x, y\ =
* 2 ! x5 3 ! x{ 4 !
l« il'i>.!ii
4/2! 8^3! 16/4!
-f _>' sinh /.
/ /3 /5 \ / /- /l \
= .*•/ + -- 1 --- 1- ... \+ y( i -| --- 1 --- \- • ••} = jc sinh/ + v cosh/.
V 3 ! 5 ! /A 2 ! 4 ! /
6. *!=*
I — tX \—tX
7. jT! = tx + (^ - i>, yi = y.
8. While the coefficients in the developments can be obtained readily, it is not
easy to recognize the functions represented by the infinite series.
6. **• 7. y.
X
Section 7
1. xy = f, p. c.,* x = y = o, i. p. 6. y =. ex, p. c., x= o, l.i.p.
2. j'2 = ex, p. c., ^r = y = o, i. p. 7. r= c, p. c., x + y = o, 1. i. p.
3. y2 = fx-\ p. c., x = y = o, i. p. 8- jog v^a^jTyj _ tan-i ? - ^ p> Ct>
4. y2 -f 2y2 = c, p. c. ^ = y = o, i. p.
5. x2 -y2 = r, p. c., x = y = o, i. p.
Section 10 f
3. *=•£ r=kg*
»"
4. *==*•+ 2;"-, i/=/.
* The abbreviations here used are: p. c. for path-curve, i.p. for invariant point,
1. i. p for locus of invariant points.
t The answers ^ivni lor the exercises of this section are not unique, since they are
particular solutions of tin- differential equations (16'). Besides, the right-hand member
of the second of these equations may be replaced by any convenient constant (see Re
mark i, $ 2) ; use of tins fact has been made in the case of Ex. 3, 4, 6.
ANSWERS 241
Section 11
I. Uf=*=£\ x = a, y = t>, p.c.;* 2 = 0, l.i. p.; x = x, y = y, z = logz,
' > y — ax, z = b, p. c.; x — y = o, l.i. p.; x = ten~l-¥- ,
dy x
+ r , z = 2, c. v. .'. x — e^ cos x, y = $ sin x, z = z.
3. Uf= x &. -\-yty- -f Z^L ; z — ax = o, z— by=o, p. c.; x = y = z = o, i. p.;
d-* dy dz
x = tan"1 — — = tan-1 — — , y = tan 1-r , z = logV^r2 +y2 + 2^, c. v.
.-. x = ez cos AT cos y, y=ez cos JT sin y, 2 = ez sin jr.
The introduction of polar coordinates reduces the group to the form of the
group appearing in Ex. I.
4. Uf= x- + y=- + xy--\ y— ax = o, xy— 2 z = b, p. c.; x = y — o, 1. i. p.
d* dy dz '
x = tan-1 2 , y = xy — 2 z, z = log vV2 +y2, c. v.
dy dz x
— bz^ = o, p. c. ; x = y = z = o, i. p. ; x — u\y y = u», z — log z, c. v.
Section 12
2. x2 -f y- — cy = o. 3. tan"1-^ = Vx* +y2 + c, spirals [p = 6 + <:].
-f ^
4. tan"1 -^ = /^ logWJ -f y2- + c, logarithmic spirals [p = ce~*~\.
5. x'2 + j- — r^r = o.
Section 21
3. xy = c*x + c, g. s.,t 4 jr2_y + I = o, s. s., y = o, p. s. for c — O ; § 25, 5. J
* The abbreviations used in the answers of \ 7 are also employed here, with the
additional one c. v. for canonical variables.
f The abbreviations used IKTC arc g. s. for general solution, s. s. for singular solu
tion, p. s. for particular solution.
J While the methods of $$ 12 and 20, especially the latter, may frequently be employed
in finding the general solution, serious practical difficulties may arise. The references
he*re given are to the places in El. Dif. Eq., where these differential equations appear
242 ANSWERS
4. ii'ty- + 2 c x + c - = o, g. s., x1 — a-y- = o, s. s., y •=. o, p. s. for c = o, § 27, 8.
5. y — c(x - c-y2, g. s., y(2~i y - 4-rJ)= o, s. s., also j = o, p. s. for c = o;
§ 26, 4
6. y2 = 2 ex + r», g. s., (32 JT* + 27^) = o, s. s., ^ = o, p. s. for c - O; § 27, 7.
7. JT 4- ^/ + ^ = o, g. s., jry- - 4 = o, s. s.; § 28, 3.
Section 24
1. The equilateral hyperbolas x* — y- = c.
2. y" = cxb. .-. y = ex, when b = a ; xy = c, when b =.— a.
3. The circles .*•- +^2 + I = <r^r.
Section 26
. ,
' d* dy' dy"
3. vw/=x&-y&-2y'&
4. £/(»>/= ^ 4- by-+ (l> _ a)yi+(t - 2 a)y"---
3-f dy dy' dy"
as exercises. Practicable methods may be found tin-re. But when the methods of
the text can be carried on), tlu-y should be employed, to obtnin practice in them.
However, the method of ^ 21 for finding the singular solution leaves nothing to be
desired. (Compare El. Dif. /•:,/. Chapter V.)
ANSWERS 243
Section 23
/o N
3. y=axe*. 4. y = x log — + c» }• 5. y = x(c^ log* + <%,).
Section 29
Section 32
3. Ar2+jj/2. 4. 5. — - 6. jr2 +/2 + 82- 2yz-2zx-2xy.
z cx—az
7- ^ _
Section 34
The group I leaves a and </ unaltered.
The group 2 leaves a unaltered.
The group 3 leaves c unaltered.
The group 4 leaves a, b and c unaltered.
Section 35
3. u=y — x, v=(x +y)(x +y + 42), or xy + yz + zx.
4. u = y + * +y~+-xz, v =y — x - yz + xz, or it =y + xz, v = .
y-X-yZ + XZ
Section 38
3. Method A, 3° applies. 4. Method B, 4°, O) applies.
5. Method B, i° applies. u = x — y, v=y — z.
+ xz
+ yz
6. Method B, 3° applies. u = ? + *Z , v=(x- -/-)(! — c3).
Section 39
3- y = log sec (x -f a) + b. 4; y2 = ax* + bx.
Section 40
5. y — log sec (j: + tf)-f //. 6. X ~c\<r. 8. <* = ax* +
J'
244 ANSWERS
Section 44
i. «. 2. 5. 3. /3. 4. /3. 5. 7. 6. 7.
Section 45*
i. JT=tan~1-^, y = tan"1 <^ + log \/
X
2. Since (UiU«)f= (/if- U»f, consider Ft/= 67i/"- Uzf=y&- and F2
5tf,/3E*|£. For these* =j, y=*.
4. JT = ^ -f y8, y = tan
= -1
+y x+y x
6. Since (UiU^f=l\f, consider Vvf= U«f, F2/= - U\f. For these
Section 49
i. x\ = x — ap, ji = y — \ ap2, p\. — p.
Vfl*/2 + IP ' \/a^~+lfl'
3- -*1! = —7* . , -> ^i = ^T-T~T» /i - 7. -m — rrs*
NOTE VI
2. 1=,,= — -, 3=
A A A A A A
A = aids - a«cn ; a{° = a^ = i, «a0 = <z3° = a4° = <z6° = o.
3. ai = — a i, «o = — 0o + ^i«:5, «;} = — #u J «i° = 02° = tf:*0 = °-
* Since multiplying its symbol by a constant does not affect the infinitesimal trans
formation of a group (Remark i, $ 2), the answers in this section are not unique. Use
is made of this fact in Ex. i, 3, 4, 6. %*
ANSWERS 245
- - I — _ a>i - _ a% . 0
o-* oy
U,f= („, + e,y + „) £. + (,„ + «,, + «,) ^.
i/6/= Ol
The groups generated l>y the infinitesimal transformations of Ex. 8 to 13 are
precisely the respective groups of Ex. I to 6.
INDEX
The numbers refer to pages.
The following abbreviations are used : dif. eq. EE differential equation ; gr. — group
infl. EE; infinitesimal ; i. u. EE invariant under ; ord. EE order ; tr. EE transformation.
Affine tr., 3, 54
Alternant, 44
of symbols of extended trs., 209
Asymptotic lines, 80
Bernoulli equation, 58
Canonical form, 26, 34, 64, 155
Canonical variables, 26, 34, 64, 156
Change of variables, 23, 33, 188
Characteristic function of infl. contact tr.,
186
Classification of two-parameter grs., 152
Commutator, 45
Complete system, 104, 106, no
equivalent, 106
Jacobian, 107, no
Contact tr., 178, 181; infl., 185
Curvature, lines of, 81
Curve of union of elements, 175, 189
Differential equation of I. ord., 189
i. u. gr., 40, 44, 45, 46, 48, 50, 52, 194, 231
Dif. eq. of 2. ord. i. u. gr., 86, 90, 134, 137,
148, 165, 236
not i. u. any gr., 206
Dif. eq. of n. order i. u. gr., 99, 101, 236
Differential invariant, 51, 88, 194
Dilatations, 185, 186
Displacements, 212
Distinct grs., 122, 123, 125
Distinct infl. trs., 7
Elements, lineal, 175
union of, 175, 194
Equivalent complete' systems, 106
Kssential parameters, 214
condition for, 226
Extended gr., 42, 84
Extended point tr., 41, 83, 180
First differential invariant, 51, 194
First integral, 191
General expression for gr. leaving dif. eq.
of i. ord. unaltered, 49
Group, i, 28, 2ii
distinct, 122, 123, 125
extended, 42, 84
generated by infl. tr,, 10, 12, 14, 30, 220
involving one parameter, i, 28
involving / parameters, 211
of contact trs., 185 .
of infl. trs., 146
property, 2
trivial, 39, 119, 196
Homogeneous dif. eq. (Boole), 93
Identical tr., 4
Independent linear partial dif. eqs., 104
Infinitesimal contact tr., 185
characteristic function of, 186
symbol of, 186
Infinitesimal tr., 6, 29, 197, 215, 218
distinct, 7
gr. generated by, TO, 12, 14, 30, 220
linearly independent, 143, 217
r-parameter gr. of, 1^6
symbol of, 8, 42, 84, 85, 218 •
Integrating factor, 37, 47, 69, 76
common to two dif. eqs., 72
two, for the same dif. eq., 48
Int> rmediary integral, 191
Invariant, 16, 31
curve, 17, 18, 31, 32
247
248
INDEX
Invariant
differential, 51, 88, 194
dif. eq., see Dif. eq.
equation, 18, 32
family of curves, 20, 22
linear partial dif. eq., 115, 118, 119, 122,
124
point, 17, 19, 31
surface, 31, 32
Inverse tr., 3, 29, 211
Involute, 70
of a circle, 70
Involution, functions in, 179
Isothermal curves, 72, 79, 203
Jacobian complete system, 107, no
Jacobi's identity, 121
Lie gr., 3
Lie's principal theorem, 225
Lineal element, 175
Linear ordinary dif. eq. of i. ord., 56, 57
of 2. ord., 92, 94, 139, 140, 173, 174
of n. ord., 102
Linear partial dif. eq. i. u. a gr., 115, 118,
119
i. u. two grs., 122, 124
Linearly independent infl. trs., 143, 217
number of, leaving dif. eq. of ord. u ~/ 2
unaltered limited, 143, 146
Lines of curvature, 81
Method of solution of
complete system, in, 113
dif. eq. of i. ord., 38, 49, 63, 66, 193
of 2. ord., 88, 134, 137, 165, 169, 193
of n. ord., 101
linear partial dif. eq., 119, 124
Minimal lines, 78
tt-times-extended gr., 84 ; tr., 83
Once-extended gr., 42 ; tr., 41
Parallel curves, 70 »
Path-curve, 4, ic, n, 17, 18, 19, 31, 67
Perspective tr., 3
Point tr., 40
extended, 41, 83, 180
Poissonian symbol, 179
Product of trs., 2
Projective tr., general, 213
Reciprocal polars, tr. by, 180, 184
Riccati equation, 52, 59, 201
Rotations, 2
r-parameter gr. of infl. trs., 146
of trs., 211, 214
Second differential invariant, 88
Separation of variables, 63
Similitudinous tr., 3
Singular solution, 66
Subgroup, 149
Symbol of extended infl. tr., 42, 84, 85
of infl. contact tr., 186
of infl. tr., 8, 218
System, complete, see Complete system
Transform of a tr., 24
Transformation
affine, 3, 54
by reciprocal polars, 180, 184
contact, 178, 181
extended, 41, 83
general projective, 213
gr. of, i, 28, 2ii
identical, 4
infl., 6, 29, 185, 197, 215, 218
inverse, 3, 29, 211
perspective, 3
point, 40
product of, 2
similitudinous, 3
Translation, 2, 53, 212
Trivial gr., 39, 119, 196
Twice-extended gr., 84; tr., 83
Two-parameter grs., classification of, 152
Two-parameter subgroups always exist,
150
Types of dif. eqs. of i. ord. i. u. given grs.,
52, 231
of 2. ord. i. u. given grs., 90, 236
of w. ord. i. u. given grs., 101, 236
Union of elements, 175, 176, 194
curve of, 175, 189
United elements, 175
Variables
canonical. 26, 34, 64, 156
change of, 23, 33, 188
separation of, 63
'
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